Disequilibrium models due to a “learning by doing” process

Rivista di matematica per le scienze economiche e sociali - Anno 16", Fascicolo 2*

D I S E Q U I L I B R I U M M O D E L S D U E T O A " L E A R N I N G BY D O I N G " P R O C E S S * CRISTIANA MAMMANA Dipartimento di Matematica Facoltd di 1ngegneria Universitd di Ancona MAURO GALLEATI Dipartimento di Metodi Quantitativi ed Economia Politica Universitd di Pescara Versione definitiva pervenuta il 08/03/94

Fluctuations and growth in a new-keynesian model are considered in three different maeroeconomic assumptions, giving rise to different mathematical models in discrete time, or maps. In particular a two-dimensional "growth model" represented by a piecewise linear endomorphism is analysed. Separating the dynamics of the slopes of straight lines issuing from the origin, from those of the points on these lines, it is shown that the dynamics of the model are governed by the dynamics of a one-dimensional endomorphism. This simplifies the study of the bifurcation values which create invariant lines on which the trajectories are divergent, and enable us to determine the basin of attraction of the absorbing region which includes regular or chaotic growth paths.

1. I n t r o d u c t i o n In this paper we analyse the generation and the persistence of growth and fluctuations in a new-keynesian model. The growth process is modelled according to a "learning by doing" process (Arrow, [1]), while fluctuations can be attributed to a nonlinear production function. Researches on this subject are usually traced back to Schumpeter [22]. According to him, innovations, i.e. shifts of the production function due to technical improvements, are the causa causans o f fluctuations and growth. After his contribution, the keynesian macro-models gained the center o f the stage but their logic was such as "to explain either fluctuations or growth, but not both" (Pasinetti [20], p. 232). The model analysed in this paper, described in the following section, resumes the Schumpeterian approach explaining fluctuations and growth. * This paper relates to the activities of the M.U.R.S.T. Group "Dinamiche Nonlineari ed Applicazioni alle Scienze Economiche e Sociali".

We note that the subject is not new. It can be considered as one more disequilibrium model of keynesian type, and several other examples can be found in the recent literature, showing complex dynamics, as the one presented in this work. As an example we recall the model studied by Dana and Malgrange [4], besides the excellent surveys on complex dynamics in economic models which can be found in [2, 3, 5, 17]. The plan of the work is as follows. In section 2 we describe three different macroeconomic models, which give rise to discrete mathematical models (non-invertible maps) of increasing complexity. The first one is a simple case, represented by a onedimensional piecewise-linear map, in which the shift of the production funcion is not taken into account. Fluctuations due tu cycles, and bounded chaotic trajectories are allowed, and occur on varying the propensity to save or the speed of adjustment of income. Some bifurcation values of this simple case are reported in the Appendix. Then, technical progress is indroduced, with a learning by doing process described by a linear function of income, and a two-dimensional map is obtained. Modelling the learning by doing process according to a nonlinear learning function (a concave function of income), a third model is described. Section 3 is devoted to the analysis of the dynamical behaviors of the second model, represented by a two-dimensional piece-wise linear map T with a non unique inverse, or endomorphism, whose dynamics are essentially governed by a one-dimensional endomorphism, named map f . Section 3.1 describes bifurcations and chaotic regimes of the one-dimensional endomorphism f , while in section 3.2 we give the corresponding dynamics of the two-dimensional endomorphism T. We see that the unique path of proportional growth, attracting in a positive domain ~D (also determined), may become repelling giving rise to two or more paths of proportional growth: lines which are cycIically visited, and on which the state of the system is growing. These lines belong to a sector of the plane bounded by critical curves, and absorbing for the points in a given domain 7). This explains how and "irrealistic" path of balanced growth may be substituted by a more realistic path of growth with fluctuations around an unstable path of constant slope, or by "irregular" (or "chaotic") trends, in which the growth is a result in the long-term behavior, while the transient shows intervals in which one of the state's variable is lowering and increasing randomly.

2. The models We deal with a closed economy without public sector. The models are strictly macro even if the microfundation arguments of Greenwald and Stiglitz [10], can be easily introduced. We assume oligopolistic markets. Therefore, prices are sticky and the adjustment process is carried out through quantities change. Saving, S, is a linear function of current income, Y, see Fig. la, S~ = sY, where0 0)" (3)

Yt+l - Y t = O(It - S t )

(see Fig. lb). In particular, since price change fails to equilibrate savings and investment, changes in income are responsible for the clearing of the market: if investment is greater (resp. smaller) than saving, income increases (resp. decreases). Substituting eq.s (1) and (2) in (3) we obtain the first model, a piecewise linear map in the variable income, Yt+1 = t(Yt), explicitely given by: Yt+1 =

(1 + r/(b - s))Yt (1 - rl(s - bo))Yt + rl(b - b o ) Z

if if

0 < Yt < Z Yt->_Z -

(4)

The assumption b > s implies that the slope (1 + r/(b- s)) of the first branch o f straight line is greater than 1. The slope of the second branch, (1 - r/(s - b0)), is nonnegative if r/(s - b0) < 1. In this case the graph of the map t is shown in Fig. 2a.

61

Y,+x

Y,+ I

/

,,.,, .-

(b)

/

0

i

y,

0

Z

y-

y,

Figure 2 Besides a repelling fixed point in the origin there is an attracting fixed point Y* whose basin of attraction is the positive interval ]0, + ~ [ . If r/(s - b0) > 1 the slope of the second branch is negative (the graph of the map t is shown in Fig. 2b), and in this case the attracting fixed point Y* may become repelling (in particular as r/and/or b increase). Some comments and bifurcation values of the simple map t in (4) are reported in the Appendix.

2.1 The second model The change of the technical progress is modelled according to a learning by doing process. We assume that a change in technical knowledge can be represented by a shift towards right of the limit Z: an increase of technical change allows the system to employ a larger amount of labor with a greater coeficient of production, increasing the level of income at which the investment curve changes its slope. Zt+~ = gt + 7Yt

(5)

where 7 is the learning coefficient (0 < 7 < 1). Of course, technical change affects the whole employment productivity; for the sake of simplicity and without any loss of generality, we role out any labor productivity increase for any X < Z. The two-dimensioal piecewise linear map T, (Zt+t, Yt+l) = T(Zt, Yt), given by (4) with g replaced by gt, and (5), will be analysed in section 3.

2.2 The third model Recent econometric evidence has shown the existence of a so-called "trend-segmented" dynamics (Rappoport and Reichlin, [21]). A possible explanation of it has been put

62

forward by Pasinetti [20]. According to him, since demand does not follow a uniform path of expansion (because of the Engel's law), periods of overinvestment will follow periods of underinvestment. In this paper we assume that technical progress is exploitable within a given "technological paradigm" (Dosi et al. [7]). In other words, within the context of a "epochal" innovation (a major innovation in Schumpeter's terminology), the rate of technical change follows a nonlinear "bell-shaped" pattern. According to this hypothesis, eq. (5) becomes: (6)

Z~+~ = Z, + 7Y~ - ~Y~'-

w h e r e 0 < 6 < 1. The two-dimensional piecewise nonlinear map T, (Zt+i, Yt+l) = T ( Z ~ , YD, given by (4) with Z replaced by Zt, and (6), is the last model described in this work, as a "natural generalization" of the second one. However, the dynamics of the third model are not considered in this paper, because they require the use of tools which are typical of a two-dimensional map with a non unique inverse, and no longer related to the bifurcations of a one-dimensional map, as is the case for the model considered in section 3. We end this section with some brief comments on the dynamics of the three models described above. For the first one, given in (4), we have that an asymptotically stable steady state may become unstable, generating fluctuating cycles; periodic or aperiodic trajectories are bounded, until a critical threshold is violated. For the second model, described by eq.s (4) and (5), fluctuations are due, as in the previous case, to the piecewise-linear production function. However, now the oscillations around a trend are always growing, due to the learning by doing process that we consider. In the last case of eq.s (4) and (6), attracting fixed points, self-sustained cyclical trends and chaotic behaviors in a bounded region of the phase plane are possible.

3. The piecewise linear two-dimensional model In this section we consider the second model, described by the following twodimensional map T: Zt+~ = Zt + 7Yt Yt+l =

(1 + q(b - s))Yt (1 - ~(s - bo))Y~ + ~(b - bo)Zt

if 0 < Y~ < Z~ if Y~ > Z t

(7)

where b > s > b0 > 0, q > 0 and 0 < 7 < < 1. T is a piecewise linear map with a "separating" line of equation Y = Z. We are interested in the dynamical behaviors of the map T in the positive quadrant, say R+2 = {(Z, Y) E R E : Z > 0, Y > 0}, as the economic variables are meaningful when nonnegative, and in the following we will consider this quadrant only. Let p = ( Z , Y) be a point of the plane, and denom as D1 (resp. D2) the portion of R 2 below or on the line Y = Z (resp. above or on the line Y = Z), we can rewrite the map as follows:

63

Pt+l = T(pt)

;

T(pt) =

( JlPt J2Pt

if if

Pt E DI pt E D2

where Jl =

1 + r/(b - s)

'

r/(b - b0)

,

1 - r](s - b0)

(8)

]

We note first that all the points of the nonnegative Z-axis are fixed points of T, and repelling (because the eigenvalues of Jt are ,Xl = 1 and ,X2 = (1 + r/(b - s)) > 1). Thus, in particular, the origin is a repelling fixed point of T, and no other fixed point of T exists outside the Z-axis. This property, together with another one (that the fixed point is a critical point, to be used in section 3.2), renders the map under examination of degenerate type. The methodology that we will follow to analyse the dynamics of T has already been used to study a particular case (the degenerate one) of the macroeconomic model in [8]. Let r p denote the point (vZ, r Y ) , then it is immediate to see that T(p) is a positively homogeneus function of degree one: PROPERTY 1. T ( r p ) = vT(p) for any v > 0 and p E R2+. If r is a half-line issuing from the origin, we denote as ri its image of rank i, that is ri = Ti(r), for i > 1. Then: PROPERTY 2. If p and q belong to the same half-line r, and q = rp with r > O, then qi = 7"pi V i > I . Property 2 states that all the points belonging to a half-line r have proportional orbits, and thus the same asymptotic behavior. However, the main consequence of property 1 is that the images, under T, of half-lines issuing from the origin are half-lines issuing from the origin. Thus, let ra be the slope of a half-line r issuing from the origin, then the sequence of slopes mi of the images ri can be obtained from a one-dimensional map. Writing m t = Y , / Z t we have, from the definition of T in (7), (1 + rl(b - s))Yt

Y~+, Zt+~

(1

if O < Yt < Zt

Z~ + 7Yt -

r](s - b o D Y , + o ( b -

bo)Zt

Zt + 7Y,

if Y~>Zt

(10)

that is, the one-dimensional map, say mt+l = f ( m t ) , is given by: (1 + rl(b - s))mt

f(mt) =

I + 7rat (1 - rl(s - bo))rat + rl(b - bo)

if 0 < m t < l if rat > 1

(11)

l .r m t

PROPERTY 3. Let m be the slope of a half-line r issuing from the origin, then the sequence of slopes mi of the images ri = T/(r), for i > 1, are given by the onedimensional map f defined in (11) (that is, rai = fi(rn), for i >_ 1).

64

It follows that we can study separately the dynamical behaviors of the slopes of the straight lines issuing from the origin, by considering the one-dimensional endomorphism f , which will be done in the next section, 3.1, and the dynamics of the points on these lines, in the plane, which will be considered in section 3.2.

3.1 The dynamics of the slopes The qualitative shape of the function f(m) defined in (11), consituted by two intersecting branches of hyperbolae, is illustrated in Fig. 3.a. It is reminescent of the one-dimensional map l of the previous section if 7 < < 1 (see Fig. 2b). f(m) is continuous, piecewise continuously differentiable, and not differentiable in the point of local maximum, m = 1.

," / /

[2:

.-"

i

.

.

.

~

m,

/I

lib

-7,

....... .

.

"\'

//!;\'

9:

o

.

/

-~,b

1"

c

',

~

6

I,

Figure 3 (a): qualitative shape of the map f given in (11). (b): graph of f(m) at s = 0 . 5 , b = 1.5,b0 = 0.01,7 = 0.1,0 = 5.2 > r/~(r/~ " 4 . 8 1 ) . The origin C) is an unstable fixed point of f , while m*, given by: m* = x/rl2(s - b~ + 47~7(b - b0) - rl(S - b0) - r/(s - b0) 27

(12)

is a fixed point of f which may be attracting or repelling. The critical points of f are denoted by el = fi§ for i >_ 0,c0 is denoted also by e. The importance of critical points in the characterization of the dynamics of one-dimensional endomorphisms, is well established nowadays, and we refer to the books by Mira [12,20] for their detailed description. In two-dimensional endomorphisms the same fundamental role is played by the critical curves (first introduced by Mira), and we shall use them in section 3.2.

65

The preimage of rank-1 of the origin O distinct from itself is denoted as O - I O - 1 and the critical point c are explicitely given by:

O-I =

;7(b-b0) O(s - bo) - 1

;

c=f(1)= l+;7(b-s)

(13)

1 +'r

For 7 < ;7(b - s), as we shall assume in tl~ following, we have c > 1 (note that our assumption is a reasonable one as the learning coefficient 7 is assumed small or very small). Then, as long as the inequality c < O - I holds, the interval I = [el, c] is an absorbing interval with basin of attraction ~D(I) =]C), O - 1 [ . About the dynamics in the absorbing interval I we have that the first bifurcation (transition of the fixed point rn" from attracting into repelling) takes place when ;7(s - bo) = 2 (in fact, - 1 < f'(rn*) < 0 for 1 < ; 7 ( s - b0) < 2, and f ' ( m ' ) < - 1 for ;7(s - b0) > 2). On varying the parameter ;7 and/or b, in the range ;7(s - b0) > 2, we observe the occurrence of chaotic dynamics in the absorbing interval 1 = [cl, c]. To fix the ideas, we shall consider the dynamics of f as a function of the parameter ;7 (in the examples of this section we have fixed s = 0.5, b = 1.5, b0 = 0.01 and 7 = 0.1, the value of ;7 is reported in the figure captions). Let ;70 = , ;71 = ( 8 - '2~ " ~ . For ;70 < ;7 < ;71 the fixed point rn* is the unique cycle of f , apart from the repelling origin. It is attracting, and its basin of attraction ~D(ra*) = ] O , O - l [ is the interval which is of interest in the applicative model, because for rn > O - 1 we have f ( m ) < 0 which is not admissible. This last statement is always true, that is, for any value of the parameters, the values of m above O - I are out of interest, giving f ( m ) < O. The complex dynamics of the map f increases as the value c approaches O - l , and, as stated above, all the trajectories in [C), O - 1 ] are bounded until the condition c < C)-l is satisfied, that is, in terms of the economic parameters: c < O-i

r

; 7 ( b - b o ) [ r l ( s - b o ) - ( 2 +7)] < ( ; 7 ( s - b o ) - l) 2.

(14)

Let ;7~ be the (unique) positive value of ;7 solution of the equation (;7(s - bo) - 1)2 ;7(b - b0)[;7(s - b0) - (2 + 7)] = 0. Then for ;71 < ;7 < ;7~ the trajectory of any point m E]O, O - I [ is attracted into the interval I = [ct, c] from which it cannot escape (I being invariant for f, f(1) = I). We shall return below on the dynamics of f inside L Let us first comment the bifurcation value ;7 = ;7~. This is the value of ;7 at which the origin becomes a snap-back repeller, SBR henceforth (for its definition cfr. [17, 18, 9]). At this value of ;7 the first homoclinic orbits of O appear (no homoclinic orbit of O exists for r / < ;7~), implying the existence of sets on which f is invariant, with chaotic dynamics in the sense of Li and Yorke [15]. We can sey that for ;7 > ;7~ the model is no longer of interest in the applicative context, because the generic point rn E [ O , O - l ] has a trajectory that will escape the positive semiaxis, becoming negative. Clearly, also in this regime there are trajectories bounded in the interval [O, O - l ] , in fact, for ;7 > r/~ a set A C [O, O - l ] , invariant for f, f(A) = A, exists, such that the restriction of f to A is homeomorphic to the shift map u : Y'~2 ~ ~"~2 being )"~2 the space of semi-infinite symbol sequences on two symbols [6, 11, 9]. That is to say, the restriction of f to A is chaotic in the sense of Li and Yorke.

66

An example of f in the regime )7 > r/~ is shown in Fig. 3b. The preimage of the origin, O - t , possesses two distinct rank-1 preimages, named O-2,~ and O-2,b, and eliminating from [O, O - l ] the points having the rank-1 image outside [O, O - l ] itself, we get two intervals: I0 = [O, O-2,~] and I1 = [O-2,b, O - l ] (see Fig. 3b). Note that 3"0 and Ii are closed disjoint intervals, such that .f(Io) = .f(IO = [ 0 , 0 - , ]

~ Io u _r~

(15)

and this is enough to prove the existence of the invariant set A with chaotic dynamics. It is obtained eliminating from [O, O - 1 ] all the points having an image of finite rank outside the interval itself, that is, removing the interval ]O-2,~,O-2,b[ and all its preimages of any rank. Although A is an uncountable set of points (a Cantor set), it is a set of Lebesgue misure zero. Thus, for )7 > r/~ almost all (in the Lebesgue misure sense) the points of the interval [O, O - 1 ] have the trajectory which escape the positive semiaxis. Let us consider now r/l < r / < r/~ and f : I = [cl, c] --+ I. As )7 increases from r/l, it is easy to prove the existence of invariant Cantor sets A C I, with chaotic dynamics (in the sense of Li and Yorke). For example, at r/= rl~ , f possess cycles of rank-k for any k E N (being N the set of the natural numbers), and in particular cycles of rank-3, which must have been born at a lower value of )7, say at )7 = r~, where r/l < ~ < r/~. Thus, after the appearance of the first cycle of period 3, for )7 > ~ (an example is shown in Fig. 4a), we can apply the condition of Li and Yorke [15]. But also before the appearance of a period-3 orbit of f , for 71 < ~ , we can prove the existence of invariant Cantor sets A C I with chaotic dynamics. Indeed, their existence is proved (for example in [6, 9]) whenever a homoclinic orbit to some repelling cycle exists, that is to say, whenever a cycle of f is a SBR. Moreover, in [9] it is proved that the SBR bifurcation value, i.e. the value of 77 at which the first homoclinic orbit of a cycle appears, is the value of )7 at which the images of the critical point c falls for the first time into the repelling cycle. For example, it is very easy to see when the fixed point rn* becomes a SBR, this is when the critical point c2 merges into rn*, say at )7 = r/~, where r/i < r/~ < r/3 (an example is shown in Fig. 4b, a chaotic trajectory at )7 = r/~ is shown in Fig. 4c). As homoclinic orbits of m* persist for )7 beyond its SBR bifurcation value, we can state that for r/~' < r / < r/~ chaotic sets in I exist. Note that at the SBR bifurcation value r/~ the map f possesses all even cycles, infinitely many, all are SBR and their homoclinic bifurcation occurred before the value r/~. Thus, f is chaotic also at values of )7 between r/1 and r/~, and in these cases we can locate the chaotic sets in a subset of 1 = [el, c] made up of two disjoint intervals because f : [cl,c3] U [c2,c] --+ [cl,e3] tO [c2,c]. An example is shown in Fig. 4d. The existence of chaotic sets can also be deduced from the statistical dynamics (eft'. the review paper by Day and Pianigiani [5]). For example, sufficient condition for the existence of a continuous ergodic invariant measure p have been proven by Lasota and Pianigiani [14] under an expansivity condition similar to the one given

67

=,.,

/

(~)

(b)

"~.t 6

...............

0-I

6

,! =*.,

(~)

m,.~

(d)

c

Ii

el

. . . . .

mf

Figure 4 8 = 0.5, b = 1.5, b0 = 0.01,3' = O. 1 (a): graph of the map f3 at r/= 4.7; (b): graph of f at r / = 4.449 ,'~ r/~'; (c) a chaotic trajectory of f at 77 = 4.449 = r/F; (d) a chaotic trajectory of f at r/= 4.25(r/1 < 77 < r/F , r/l ~ 4.081).

in (15), that is, under the existence of two compact disjoint sets A and B such that fk(A) fq f~(B) 3 A U B for some k E N. Now we note that if f possesses a cycle of period q, q > 1, which is a SBR (as it occurs for example at least for r/ > r/~'), then an integer k > q and two closed disjoint intervals U0 and Ul exist such that fk(Uo) N f~(Ul) D Uo U U1 (see [9]). Thus for 77 > r/F a continuous ergodic invariant measure # exists. We remark that the chaotic sets may have Lebesgue misure zero (as it occurs for example for 77 > r/~), while we are interested in the knowledge of values of r/ at which the chaotic sets are "big enough" in a measure theoretic sense. To the scope, we recall that a sufficient condition for the existence of an absolutely continuous invariant measure/~ has been proven by Lasota and Yorke [13] under the assumption that the map is expansive on a trapping interval (a piecewise C 2 map f : I ~ I is called expansive on an interval I if the derivative f~ satisfies Jf'[ > 8 > 1 almost

68

everywhere in I). We note that if f is expansive, then all its cycles are repelling. Moreover the result holds also in the weaker condition that fk is expansive for some k E N. For our map f under examination, f : I = [cl, c] ---* I, the slope f~(rn) is much higher than 1 for any m E [el, 1[ for rlz < r / < r]~, and also the slope of the second branch of f is very steep. For example, a sufficient condition for If'l > 6 > 1 for any m E] 1, c] is get when the parameter 3' is small enough, as a value rb, exists, r]~' < r], < r]~, such that f'(c) < - 1 for 7/ > rb, so that f is expansive in I for r/8 < r] < r]~. More generally, assuming 7 < 0.1 (as is reasonable), a value f], exists, rll < f], < r]~, such [(fk)' I > 6 > 1 for any ra E I, m :~ 1, for some k > 2. That is, fk is expansive in I, and an absolutely continuous invariant measure/~ exists for rls < r] < r]~ for fk in 1. In such cases, the extended theorem by Li and Yorke [16] also applies, proving the existence of ergodic absolutely continuous invariant measures.

3.2 The dynamics in R 2+ As we have seen, the two-dimensional map T maps half-lines from the origin into half-lines from the origin. If ri is the half-line from the origin of equation Y = miZ, then ri is mapped into the half-line ri+l = T(ri) of slope mi+l given by rni+l = f(rai) (i.e. of equation Y = mi+lZ). We call so deduce geometrical properties of T of global character (as absorbing regions and their basins of attraction), from analogous properties of the one-dimensional map f . It follows that the critical curve of T of rank-1 is the half-line, say LC, image of the half-line with slope rn = 1 (i.e. of equation Y = Z, which is called line LC'_I). That is, LC is the half-line with slope c = f(1). And in general, the critical half-lines LCi issuing from the origin have slopes ci, for any i > 0. In particular LCI has slope el. Thus, in the positive quadrant of the plane, the sector, or cone, say Cx, bounded by the half-lines L C and LCI, is an absorbing region (see Fig. 5). Its basin of attraction is the cone 7) bounded by the Z-axis and the half-line which is its preimage of rank-I, that is the half-line, say Z_l, with slope (~)-lNote that we could as well deduce the geometrical properties of T from the direct analysis of its critical curves (following the pioneering examples in [12]). In fact, from the definition of T in (8) it follows immediately that the locus of points of the positive quadrant of the plane in which T is not differentiable is the half-line LC_l of equation Y = Z, and that the critical curve of rank-I, in the Julia-Fatou sens (locus of points having at least two coincident preimages of rank-l) is the halfline L C = T(LC_I) (the critical lines of rank-(/+ 1) are defined as usual with the images: LCi = TI+I(LC-I) for i > 0, [12]). It follows that as long as LC2 is an half-line belonging to the cone Cz bounded by LC' and LCI, this cone is an absorbing region. Moreover, being the Z-axis locus of fixed points, this axis clearly belongs to the frontier of the basin of attraction 29 of CI, and the other half-line of the frontier is the rank-1 preimage of the fixed points, called Z_l. However, the

69

y'

z

i

Figure 5 deduction of these properties from the one-dimensional map f is easier, and it offers the advantage of a direct visualization of the dynamics of the slopes of the half-lines ri, impossible otherwise. We note also that in our map T the critical curve L C separates the plane in two regions, the points of which have zero and two distinct preimages of rank-one, but all the critical curves of T, of any rank, pass through a fixed point of T belonging to L G (here the origin) and this renders the map of degenerate type. The fixed point m* of f corresponds to (that is, gives the slope of) a unit half-line of T (i.e. which is mapped into itself), which, in its turn, is a repelling eigenvector of 3"2 corresponding to the positive eigenvalue (it is easy to see, in fact, that J2 possesses two real eigenvalues, say A_ and A+, with A_ < 0 and A§ > 1). The points of R 2 outside ~ have trajectories which are economically unmeaningful because they have points below the Z-axis, as the rank-1 image of a point p E R 2 \ ~ has negative Y-coordinate. We can so state that the region of the plane which is interesting is only the sector "D. For any point p E 29 we have that either it is a point of the cone Ct or an integer n exists such that the image of p of rank-n belongs to tTt (i.e. T~(p) E Ct), and from Ct the trajectories cannot escape (being T(Cx) C (?t and Ct is absorbing). Now, let us look for the dynamics inside the cone

C~. We recall first that the informational content of the map f with respect to T is of global character, as the dynamical behavior of rn (slope of a half-line r) under f gives the dynamical behavior of the sequences of the half-lines r i = T / ( r ) , but not that of the points on the ~ s undetr T. As an example, to a periodic orbit of the map f of period k, say {mr,m2,... ,rnk}, there correspond a set of k half-lines, say {rl, r 2 , . . . , rk} (with slopes ml, rn2,..., rnk, respectively), which are cyclically invariant under the map T, that is, T(ri) = ri+l and Tk(rj) = rj for j = 1 , . . . , k. Thus, if Pl belongs to rl, its image under T, p2, belongs to 7"2,and so on; the image of Pl of rank-k belongs to rl, and iteratively, Tnk(pl) E ri for any n > 1 (and in general T"~(pj) E rj for any n > 1,j = 1,... ,k), but the orbit o f p l is not periodic

70

on the half-lines {rl, r 2 , . . . , rk} whereas it is divergent. To see this we rewrite the first equation, in (8), of the map T as: Zt+l = (1 + 7mt)Zt

(16)

so that (1 + 7m~) > 1 for any rat > 0 implies Zt ~ +oo as t ~ +oo (assuming an initial value Z0 > 0 as is the case of interest here). Now, from Yt = rntZt, for values of the parameters such that e < O - I we have that ms is a sequence bounded in the absorbing interval [cl, c] as t ~ +oo, and this together with Zt ~ +oo as t ~ +oo, implies that Yt must also be divergent. We can conclude with the following property: PROPERTY 4. The trajectories of the map T in D C R 2+are always divergent. However, the qualitative behavior of the trajectories depends on the value of the parameters. Let p be a point in D and rn be the slope of the half-line from the origin through p. Then, for any i > 1 the images Pi = T/(.P) belong to half-lines of slopes mi = f i ( m ) . Thus the dynamics of the sequence {mi} give the dynamical behavior of the half-lines to which Pi belongs, and on these lines Pi is divergent as i ~ +oo. We have: PROPERTY 5. If 1 < ~7(s - bo) < 2 then there exist a unique "path of proportional growth" (the half-line of slope ra*) which is attracting in the sector D. In fact, we have seen that if 1 < r}(s - b0) < 2 the fixed point m* of f is attracting in the interval I = [cl,c] with basin of attraction ( O , O - 1 ) . Thus, any half-line r of the sector D has images ri = T/(r) which are convergent to the eigendirection r* (half-line with slope rn*) of d'2, corresponding to the eigenvalue )~+ > 1. In particular, if we consider a point/90 E r*, then its images belong to r* and are given by Pi = Ti(,po) = )~P0. If P0 E D, and P0 ~ r*, then its images Pi approach the line r * and, as i --* +o0,pi tend towards the point at infinity on r* (see Fig. 6a). PROPERTY 6. I f ~7(s - bo) > 2 and condition c < O - t holds, then the images pi of a point p E D are divergent in !r 2+with regular or chaotic transient, depending on the corresponding sequence of slopes ml. For corresponding sequence of slopes mi we mean, as before, the images of the slope rn of the half-line from the origin through the point p E D. Thus, when the dynamics of the map f(rn) are regular, for example an attracting k-cycle exists, then the generic r is convergent to a set of k cyclically invariant half-lines, and the Pi will approach these half-lines (see Fig. 6b); when the dynamics of the map f(rn) are chaotic on some intervals, then the points Pi are divergent with chaotic transients. Two examples are illustrated in the figures 6c and 6d. Note that to all the repelling cycles of f correspond cyclically invariant half-lines of T on which the points are divergent with proportional growth.

71

/ i

N

!

/

(~)

?:i: ~,

,:-~

-.

.... -,

;: i

.

ii . .

.

"

"

"

"

.ii-.":

"

"

9

::

:i)

(~)

. Ca) )

Figure 6 Trajectories of the map T given in (9), in the region [0,100] • [0,100] of the (Z, Y) plane, s = 0.5, b0 = 0.01, b = 1.5, 7 = 0.001 and r/= 3.5 in (a); r/= 4.082 in (b); r/= 4.2 in (c); r1 = 4.5 in (d).

4.

C o n c l u s i o n s

Several linear models in the economic literature deals with the existence of a unique path of balanced growth (recall the Frobenius-Perron path). If a nonlinearity is introduced in these systems, it may be that nonlinear effects will come to play a dominat role, as it is the case in the two-dimensional new keynesian model considered in section 3. The growth process is not lost, whereas it is substituted by fluctuating, or even chaotic, growth paths, which is perhaps more adequate, in several context, to

72

interpret the economic behaviour. The two-dimensional map considered in section 3 is an endomorphism, and its being of degenerate type facilitate the analysis of the dynamical behaviors. In fact, this enable us to separate the dynamics of the slopes of the stright lines issuing from the origin from those of the points in these lines. This can be used to give a complete description of the bifurcation mechanism (related to that of a one-dimensional endomorphism) which creates cyclical lines on which the trajectories are divergent. Moreover, the domain of attraction, in the positive quadrant of the plane, of the absorbing region which includes regular or chaotic growth paths, has been determined.

Appendix This appendix is devoted to some short comments on the dynamics of the piecewise linear map Yt+1 = ~(Yt) given in eq. (4), which we rewrite for reason of convenience, -s))Yt

Yt+l =

if if

f(l+r/(b (1

- rl(s - b o ) ) Y t + rl(b - b o ) Z

0 s > bo > 0 and Z is a constant value, Z >_ 1. As already proven in section 2, values of the parameters in the regime rl(s - b0) > 1 are those which interest us in order to determine when attracting cycles and chaotic dynamics occur (in this regime t is a particular "trent-map", whose shape is shown in Fig. 2b). Thus we shall assume rl(s - b0) > 1. The critical point of the map t is the point c = t ( Z ) = (1 +r/(b - s ) ) Z . T h e critical point of t of rank-(/+ l) is given by cl = t i + l ( Z ) , for i > 0. It is easy to see that the fixed point Y* of t: y,

= (b - b o ) Z

(s - b0)

(A.2)

is attracting if r/(s - b0) < 2. For 1 < r/(s - b0) < 2 the basin of attraction of Y* is the open interval :D(Y*)=]O, C)-l[, where C) is the repelling fixed point origin and C)-l the preimage of rank-1 of the origin distinct from itself, which is given by rl(b - b o ) Z C)-~

= ( r / ( s - bo) -

1)

For ~7(s - b0) > 2 the fixed point Y* is repelling, and in this regime we have that I = [cl,c] is an absorbing interval as long as c < C)-l, with basin of attraction the open interval 7)(/) =]O, O - 1 [ . Clearly, for any value of the parameters, the points outside the closed interval [C), C)-l] have trajectories divergent to - o r At the bifurcation value r/(s - bo) = 2, the interval I = [cl, c] (where cl = c_1 = Z) is filled with peridic orbits of period two, and the basin of attraction of I is always 7:)(/) =] (~)O-1[. This means that any point in the interval ] O , C)-1[ is either periodic

73

or eventually periodic (i.e. it will fall into a 2-cycle of the interval I in a finite number of iterations of the map t). For r/(s - b0) > 2, as stated above, all the trajectories in ] O , O - l [ are bounded and attracted from the absorbing interval I, as long as the inequality c < O - l holds, that is, in terms of the economic parameters: e < O-1

r162 r/2(b -

s)(s

-

bo) -

2r/(b - s) - 1 < 0

(A.4)

In the regime r/(s - b0) > 2 all the existing periodic orbits of the map t are repelling, because the slopes of the two branches of straight lines consituting t are both greater than 1 in absolute value. That is to say, the map t is expansive. We know that an expansive map possesses chaotic sets, and in this case (a piecewiselinear map), we can say something more. Let us consider the dynamics o f t as a function of the parameter )7. Let rh be the value of r/at which the fixed point Y* becomes repelling (rh = o-;-s~o)), 2 . r/[' the value of r/at which the first homoclinic orbits of the fixed point Y" appear (i.e. the SBR bifurcation value of Y ' , which occurs when the critical point c2 merges into the fixed point Y*), r/~ the value of r/at which the first homoclinic orbits of the fixed point origin appear (i.e. the SBR bifurcation value of O , which occurs when the critical point cl merges into the fixed point O , or, equivalently, when c = O - 1 , thus, from (A.4), r/~ is the positive root of the equation r/2(b - s ) ( s - b o ) - 2~7(b - s) - 1 = 0). The three bifurcation values rll < r/~ < r/~ characterize different regimes. As stated above, the map t is chaotic for )1 > r/~, but only in the range rh < )7 < r/~, t is chaotic in some intervals. We recall that a map is chaotic in a closed invariant set ,4 if the trajectory of any point o f , 4 is either periodic or eventually periodic or aperiodic and dense in .,4. For r / > r/~, the interval [ O , O - l ] is not mapped into itself by t; inside it, a Cantor set of Lebesgue measure zero exists on which t is invariant and chaotic, but almost all the points have a trajectory which escape that interval and diverges to - o o . Thus we have that the chaotic regimes of interest in the applications are those corresponding to rh < r/ _< r/~. At these value of )7, ergodic absolutely continuous invariant measures exist. In particular, for rh < r / < r/~', t is chaotic on a set .,4 made up of two disjoint intervals, ,4 = -Tot2 I1, where Io = [cl, c3] and 11 = [c2, c] (each of which is invariant and chaotic for the map t2), while for r/~ < )7 < r/~,t is chaotic on the interval l = [cl, el, I C [ O , O - I ] (the equality holds only at the bifurcation value r/= r/~). We remark that for rh < )7 < r/~' no periodic orbit of t of odd period can exist, apart from the fixed points, because all the other cyles have a period which is a power of 2. Some examples are shown in Fig. 7 In Fig. 7a, t is chaotic in the union of the two disjoint intervals [cl, c3] and [c2, e] (i.e. we are in the regime r/l < )7 < r/~). This set is absorbing and attracts all the points of ] O , O - I [ apart from the repelling fixed point Y ' . At )7 = r/~', as the example in Fig. 7b, t is chaotic in the interval I = [el, c], however at this bifurcation value we have I = l o t 3 l l(as c2 = c3 = Y*), so that the map t 2 is also invariant and chaotic in the two intervals -To and Ii (this is no longer true after bifurcation). Fig. 7c shows an example in the regime r/~ < r / < ~7~ (at a value of )7 near the bifurcation value r/~); t (as well as the map t 2) is chaotic in the interval I = [cl, el.

74

/\ i

i

i

I /

/

,

i

I

b)

(~) I

, I

. . . . . . . . .

i

/o,

:/

\]I

1

1

I)

Figure 7 Chaotic trajectories of the map t in (A.1) represented in the region [0, 15] x [0, 15] of the (Yt, Yt+l) plane, s = 0.5, Z, = 1,b0 = 0.01,b = 1.5 and 9 = 4.18 in (a); r/= 4.285 ~_ r/~" in (b); r/= 4.5 in (c).

REFERENCES [1] ARROW K. J., The Economic Implications of Learning by Doing, Rewiew of Economic Studies, 29, 155-173, 1962. [2] BOLDRIN M., WOODFORD M., Equilibrium Models Displaying Endogenous Fluctuations and Chaos: A Survey, in Cycles and Chaos in Economic Equilibrium, J. Benhabib (ed.), Princeton University Press, 1992. [3] CUGNO F., MONTRUCCHIO L., Some New Techniques for Modelling Nonlinear Economic Fluctuations: A Brief Survey, in Nonlinear Models of Fluctuating Growth, R. Goodwin, M. Kruger, A. Vercelli (ed.s), Springer Verlag, 1984. [4] DANA R. A, MALGRANGE P., The Dynamics of a Discrete Version of a Growth Cycle Model, in Analysing the Structure of Economic Models, J. P. Ancot (ed.), Martinus Nijhoff, The Hague, 1984, 205-222. [5] DAY R., PIANIGIANI G, Statistical Dynamics and Economics, Journal of Economic Behavior and Organization, 16, 1991, 37-83. [6] DEVANEYR. L., An Introduction to Chaotic Dynamical Sistems, Addison Wesley, New-York, 1987. [7] DosI G., FREEMAN C., NELSON R. R., SILVERBERG G., SOETE L., Technical Change and Economic Theory, Frances Pinter, London, 1988. [8] GARDINI L., On the Global Bifurcation of Two-Dimensional Endomorphisms by Use of Critical Lines, Nonlinear Analysis, T.M.&A., 18(4), 1992, 361-399. [9] GARDINI L., Homoclinic Bifurcations in n-Dimensional Endomorphisms Due to Expanding Periodic Points, Nonlinear analysis, T.M.&A. (to appear), 1992. [10] GREENWALDB. C., STIGLITZ J. E., Financial Market Imperfections and Business Cycles, Stanford University, mimeo, 1991. [11] GUCKENHEIMER, HOLMES P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New-York, 1983.

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[12] GUMOWSKI I., MIRA C., Dynamique Chaotique, Cepadues Editions Toulouse, 1980. [13] LASOTA A., JORKE J. A., On the Existence of Invariant Measures for Piecewise Monotonic Transformations, Transactions of the American Mathematical Society 186, 1973, 481-488. [ 14] LASOTA A., PIANIGIANIG., Invariant Measures on Topological Spaces, Bollettino UMI, 14B(5), 1977, 592-603. [15] LI T-Y, YORKE J., Period Three Implies Chaos, American Math. Monthly, 82(10), 1975, 985-992. [16] LI T-Y, YORKE J., Ergodic Transformations From an Interval Into Itself Transactions of the American Mathematical Society, 235, 1978, 183-192. [17] LORENZ H. W., Nonlinear Dynamical Economics and Chaotic Motion, SpringerVerlag, New-York, 1989. [18] MAROTrO J. R., Snap-Back Repellers Imply Chaos in R '~, J. Math. Analysis Applic. 63, 1978, 199-223. [19] MIRA C., Chaotic Dynamics, World Scientific, Singapore, 1987. [20] PASINETrl L., Structural Change and Economic Growth, Cambridge University Press, Cambridge, 1981. [21] RAPPORT P., REICHLIN L., Segmented trends and non-stationary time series, European Economic Institute, 1987, 87-319. [22] SCHUMPETERJ. A., The Theory of Economic Development, Cambridge University Press, Cambridge, 1934.

Modelli di disequilibrio dovuti ad un processo di "learning by doing"

SOMMARIO Si considerino fluttuazioni e crescita in un modello neo-keynesiano in tre diverse ipotesi macroeconomiche, che darmo origine a diversi modelli matematici in tempo discreto, o mappe. In particolare si analizza un "modello di crescita" bidimensionale, rappresentato da un endomorfismo lineare a tratti. Separando il comportamento dinamico delle pendenze di rette che escono dall'origine, dalla dinamica dei punti su tali rette, si mostra che il comportamento dinamico del modello ~ governato dalla dinamica di un endomorfismo unidimensionale. Cib semplifica 1o studio dei valori di biforcazione che creano rette invarianti, su cui le traiettorie sono divergenti, e consente di determinate il bacino di attrazione di regioni assorbenti che contengono i cammini di crescita, regolari o caotici.

76