Competitive chaos

JOURNAL

OF ECONOMIC

40, 13-25 (1986)

THEORY

Competitive RAYMOND

Chaos* DENECKERE

Department of Managerial Economics and Decision Sciences, Northwestern University, Evanston, Illinois 60201

AND STEVE PELIKAN

University Received

Department of Mathematics, of Cincinnati, Cincinnati, Ohio 45221

September

4, 1985; revised

January

13, 1986

What kinds of dynamic behavior can optimal trajectories in deterministic growth models display? This paper presents examples of economies that have stable equilibrium cycles in consumption, capital, and prices of arbitrary period, as well as of economies that have chaotic equilibrium paths. Some necessary and sufficient conditions for these phenomena to occur are discussed. Journal of Economic Literature Classilication Numbers: 021, 023. 111. ‘i‘! 1986 Academic Press. Inc.

1. INTRODUCTION What kinds of behavior can the optimal trajectories in a growth model display? The model we study is a standard deterministic growth model. While analytically simple, it imposes the strongest possible restrictions on equilibrium paths of capital accumulation in competitive economies. Any phenomenon observed in simple models of this kind will, a fortiori, be present in more complex models, such as models incorporating several types of infinitely lived agents, or models which have a mixture of finitely lived and infinitely lived agents (hybrid overlapping-generations models). In the standard one-sector growth model with a single capital good it is * This research was partly undertaken while the authors were visiting the Institute for Mathematics and its Applications at the University of Minnesota. Ray Deneckere’s research was supported in part by the National Science Foundation Grant SES-854701; Steve Pelikan’s research was funded in part by National Science Foundation Grant DMS-8542778. We would like to thank J. Benhabib, W. Brock, R. Manuelli, and G. Sell for helpful conversations and an anonymous referee whose useful comments and suggestions greatly improved the exposition of this paper. Any errors are, of course, solely our responsibility.

13 0022-0531/86

$3.00

Copyright 0 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.

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well known that capital trajectories converge monotonically to a steady state (Dechert and Nishimura [6]). Similarly, in the multisector growth model (with one or several capital goods) convergence of optimal paths occurs if the discount factor is sulliciently high (Scheinkman [14], Brock and Scheinkman [3]). The example by Sutherland [15] shows that this so-called turnpike property need not hold when the discount factor is small enough. In fact, his example has optimal cycles of period two. Subsequently, Benhabib and Nishimura [l] gave suflicient conditions for the existence of two-cycles. In this paper, we show that the dynamics of optimal growth paths can be arbitrarily complex. We present examples of economies (two-sector growth models with a single capital good) that have stable equilibrium cycles in consumption, capital, and prices of any periodicity, as well as of economies that have chaotic equilibrium paths. Some necessary conditions (in terms of technology and preferences) for these phenomena to occur are discussed. We also provide sufficient conditions for chaotic dynamics to occur. The existence of growth problems with “chaotic” optimal paths (and chaotic price paths) has implications concerning the assumption of perfect foresight. In an environment which is sufhciently regular (e.g., periodic) learning mechanisms of the sort investigated by Grandmont [S] can account for foresight. It seems unlikely that such mechanisms exist in cases where the environment is highly erratic. It would appear then, that in such cases, optimal behavior would not be observed in practice.

2. The optimal

growth problem

THE

MODEL

we consider is the following:

(P) Given fl E (0, 1) and x0 E R, maximize to (x,, xi+ I) E D for j = 0, 1, 2 ,..., and x0 = x.

c,$

/VV(xi, xi+ 1) subject

Here D is a compact convex domain in lR2 and the return function real-valued, defined on D. We assume throughout:

V is

(Al) D= {(x, y): O x on the interior of D. (A.2) v,,v22-

V is Cz on D, with

V, > 0,

I’, ~0,

V,, ~0,

V22 ~0,

v:,>o.

Problem (P) and (A.l), (A.2) arise, for example, from a two-sector model of optimal growth: maximize C,“=0 p’u(c,) with respect to {I:, If, xi, x:}

COMPETITIVE

15

CHAOS

subject to the constraints c,
1.

h&x)

is jointly

continuous

in

(b,

x)

on

(0, 1) x [0, L].

We search for the function h& * ) directly as follows: set W,(x) = max~;?Y&/?‘V(x,,x,+i) subject to (xi,xj+,)~D and x,,=x. Then IV,(.) satisfies W,(x)= max {W, y)+BWs(y)}. (1) O<.P
In this secton we first show that when /? is near 1 or 0, hs(. ) has a globally attractive fixed point. We then give an example of a growth

16

DENECKEREAND

problem which has graphical explanation It can be shown follows, e.g., from following somewhat

PELIKAN

a stable cycle of period 2. We end the section with a of conditions necessary for complicated dynamics. that h& .) converges uniformly to 0 as p -+ 0 (this Boldrin and Montrucchio [2], fact 1). In fact, the stronger result is true:

PROPOSITION 2. There exists a p > 0 so that if /I < /? the policy function of (P) is identically zero.

ProoJ Let P=mino,,.,,,, { ( V2(x, y)(/V,( y, O)}. It is easily checked that the function W,(x) = V(x, 0) + (b/( 1 - 8)) V(0, 0) solves the functional equation (1) and hence that h,(x) = 0, for B d p. i

Next, we consider (P) for p near 1. LEMMA 3. There exists fi < 1 so that zf/? > /?, h, has no periodic points of period n > 2.

Proof: Because h,(. ) is continuous, we may apply Sarkovksii’s theorem [13] and conclude that if hp(. ) has a point of minimum period k > 2, then it also has a point of period 2. Consequently, it is enough to show that there is a fl< 1 so that for j? > j?, the function h, has no points of period 2. Suppose that p is such that hp(x) = y, h,(y) = x and y > x. We will show that b is bounded away from 1. We can immediately rule out the case where x = 0 and t(0) = 0. By considering feasible variations on the path (x, y, X, y,...,) it is easy to prove that x and y must satisfy:

V,(x, Y) + Wl(Y.

xl 3 0

Vz(Y, x)+Bvl(x,Y)Go

( =0

if

y < t(x)),

(=0

if

x>O).

(2)

We claim that there exists /?, < 1 and E > 0 so that any pair (x, y) E D which satisfies (2) also satisfies )x-y ) > F. To see this, expand (2) in a Taylor series at (X, X) with X = (x + y)/2, retaining the constant and linear terms. Subtracting the resulting expressions yields: o~(y-x)c~,,+P~,~-(~+B)

~1*1(~,~)+4l~-Y/).

(3)

The strict concavity of V implies there exists a j, < 1 so that PVi, + V,, (1 -t-B) V,,O. If Ix-y/ were arbitrarily small, (3) would be violated. The strict concavity of V implies there exists a S > 0 with V qq (

1

>[V(x,y)+V(y,x)1/2+6

for all (x, y) E D with [x - yl > E. It then immediately

follows that there

COMPETITIVE

CHAOS

17

exists a f12< 1 so that for all /I > fi2 the path (x, X, X,...,) dominates the path (x, y, x, y ,... ). Letting fi = max((f,, fiz), the theorem is proven. 1 In fact, stronger results than Lemma 3 are known. Define x* = argmax I/(x, x). Under the assumption that x* > 0 Scheinkman [ 141 and Brock and Scheinkman [3] have shown that there exists a fl< 1 such that for all b > fl the sequence { hj,(x)},T , converges to a (unique) fixed point of h, for all x E (0, 11. It can be shown (Deneckere and Pelikan [7]) that this so-called “turnpike theorem” follows from Lemma 3 and a result of Coppel [S], which states: Iff is a continuous map of a compact interval to itself that has no cycle of period 2, the sequence (f’(x)};?!, converges to a fixed point off for every x in the interval. For fl E (/I, 8) it is possible for the policy function h, to have more complicated dynamics. Consider the function V(x,y)=

-~y+fg)(l-~x*)y-~y*+~x 1

-+x--m

49 (1 - $x2)2.

I/ is strictly concave on D = [0, l] x [0, 11, and satisfies the monotonicity conditions of (A.2). It is easy to check that for fi B 0.683, h, is identically equal to 1. Thus, for this example, fi < 0.683. Furthermore, /? can be calculated to be: p = 0.004. For intermediate values of 8, h,( . ) possesses a unique fixed point, which decreases monotonically in fi. This fixed point is stable for b values in (0.253, 0.6831. As p crosses 0.253, the slope of the policy function passes through - 1. The stable steady state loses stability and gives birth to a stable orbit of period 2. Benhabib and Nishimura [ I] derive conditions on V under which such a period doubling bifurcation takes place. For /3 = 4 we can find this orbit by iteration, as hp(x) = 1 - $x2 (this can be verified by showing that at b = 4, W,(x) = 9x - 4.~’ solves the functional equation (1)). In fact, for this value of the discount factor, the periodic orbit is globally attractive (it attracts all points in the unit interval except the fixed point of h,)! Observe that in this example V,, = -$$,x < 0. Then the next result implies that all the periodic orbits of the example have either period one or period two and that each trajectory is asymptotic to one of them. ~OPOSITION 5. (i) Suppose that ( x, y) is a point on the graph of the policy junction h, and that I/,*(x, y) > 0 (< 0, and that y < t(x)). Then h is nondecreasing (nonincreasing) at x. If (x, y) is in the interior of D then h is increasing (decreasing) at x.

(ii) According[y, if V,, > 0 on D, h, has no cycle of period n 2 2, and for every x in [0, L], hi(x) converges monotonically to a steady state, If

18

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V,, < 0 on D and t(x) = L for all x in [0, L] (or h,(x) < t(x) for a/l x), all periodic orbits of h, have period one or two, and each trajectory h$(x) is asymptotic to some periodic orbit.

(i) See Benhabib and Nishimura [l]. (ii) According to (i), the conditions of (ii) with V,, < 0 imply that the policy function h, is nonincreasing for all values of fi. In particular, then, h2(x) = h(h(x)) is nondecreasing in X. The trajectory (hZ,“(x)),“= 1 is therefore monotone and converges to a fixed point of hg (that is, a periodic point of h, of period one or two). If V,, > 0, a similar argument shows that every trajectory converges monotonically to a steady state. 1 Proof

It is possible to give a graphical interpretation of these results. Bellman’s principle reduces the problem (P) to a two-period problem: maximize V(x, y) + /?W(y) subject to (x, y)~ D. Then (P) can be rewritten as: maximize u + PI+‘(y) subject to (x, y) E D and u < V(x, y). For fixed x, u = V(x, y) determines a concave budget constraint in (y, U) space (Fig. 1). The fact that V, > 0 says that this constraint moves out as x increases. The fact that V,, < 0 implies that the (absolute value of the) slope of the budget frontier increases as x increases. To solve the maximization problem we select (ignoring the constraint (x, y) ED for ease of exposition), for a given x, that value of y = h,(x) for which the budget constraint and indifference curves are tangent. Now, if X’ >x and y’= h,(x’) is the value for which tangency occurs for x’, the fact that indifference curves are vertically parallel implies that y’
+ fl W ( y 1 = constonl

FIG. 1. A graphical solution of the one-period when VI2 < 0, the policy function h is decreasing.

programming Here I < x’.

problem

which

shows

that.

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CHAOS

19

To interpret the condition VI2 < 0, let us return to the two-sector onecapital good model introduced in Section 2, and let us assume that utility is linear in consumption: u( c,) = c,. Under this assumption, and assuming that f( ., .) and g( ., .) exhibit constant returns to scale, Benhabib and Nishimura [l] proved that V,, < 0 ( > 0) whenever the consumption goods sector is more (less) capital intensive than the investment sector. When f= g, consumption can be transformed into future capital on a one-to-one basis, i.e., we really have a one-sector growth model. In this context, Dechert and Nishimura [6] have shown that when u(. ) is strictly concave, all trajectories converge monotonically to some steady state, even if the production function is not required to be concave. In fact, since in this case V( X, J,) = u( 4(.x) - y), their result follows immediately from Proposition 5, as long as the production function d(x) =f(.u, L) is strictly monotone in X. A one-sector growth model can thus only display, in our terminology, simple dynamics.

4. COMPLEX

DYNAMICS

In the previous section we saw that a necessary condition for the existence of solutions of (P) which have periodicity greater than two is that I’,, not be of uniform sign on D, at least as long as the policy function remains in the interior of D for x in (0, L). In this section we demonstrate by example that when V,, does take on both positive and negative values in D, the trajectories of the policy function can have very complicated (chaotic) behavior. A number of different definitions have been proposed for the term “chaos” in the context of iterated maps. Many of these notions may not be appropriate in the context of economic models such as ours. For instance, the paper “Period Three Implies Chaos” by Li and Yorke [ll] shows that if a map of an interval to itself has a point of period 3, then there is an uncountable set of initial conditions which give rise to chaotic trajectories (i.e., trajectories which are aperiodic and not asymptotic to a periodic trajectory). This does not preclude the case where for a set of initial conditions of probability 1 (i.e., for a set of full Lebesgue measure in the interval) trajectories converge to the stable period three orbit. The notion of complex dynamics which is advocated in the literature (see, e.g., Grandmont [9]), and which we describe below, is designed to eliminate the possibility that there are some trajectories which behave in a complicated manner, but that almost all trajectories show a simpler behavior. Given a map h of an interval I to itself, we say that a probability measure /J on I is h-invariant if p(h-‘(A)) = p(A) for each measurable set A. One possible way to model complex dynamics is to assume the existence

20

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of an ergodic h-invariant probability measure ,u that is absolutely continuous with respect to the Lebesgue measure. This implies that for ,ualmost every X, hence a set of positive Lebesgue measure, the empirical distributions p”(x) (with p”(x) = l/(n + 1) c,“=O &h’(x)), where 6(y) is the measure that assigns probability one to (y}) converge weakly to the measure ,a. Thus trajectories starting in the support of p will look rather erratic since they eventually Ii11 almost surely the whole support. For this reason, we will say that the map h has complicated dynamics. Another definition of complex dynamics often used in this area is that the map h generates orbits which are very sensitive to small variations in initial conditions. Such chaotic behavior has been observed, e.g., in turbulent flows. If h has an ergodic absolutely continuous measure n, and log (h’(x)/ is p-integrable, then lim A -2’ log 1h’(h’(x)( = 5 log 1h’(x)( dp(x), n+m n /=O p-almost everywhere. The left side of (4) is called the Lyapounov exponent of h at x. It gives an average (linearized) contraction and expansion rate along the orbit h’(x). (4) says that Lyapounov exponents exist and are independent of x, p-almost everywhere. If the Lyapounov exponent b = J log 1h’(x)/ dp(x) is positive (which need not be true in general but will be the case in the example we present below), then (4) says, roughly, that I(d/dx) h”(x)/ > en’ >> 0 for a set of x having positive Lebesgue measure and n sufficiently large. Consider what this means: for a large set of nearby initial conditions x and y we will have, at least to a linear approximation, that the trajectories through x and y diverge “on average” at the exponential rate 6. Such sensitive dependence on initial conditions, in the context of optimal growth models, means that extremely accurate knowledge of initial capital is required in order that the trajectory generated by the policy function even approximate the optimal trajectory on finite (but large) time intervals! Recall that if, as in the end of Section 3, we imagine that there are two sectors producing utility of consumption and future capital, then the sign of I’,, expresses the relative capital intensity of these sectors. The fact that V,, takes both positive and negative values indicates that capital-intensity reversals take place. To see geometrically how these reversals can result in more complicated dynamics for the policy function, consider Fig. 2. Here we have assumed that I’,, = d(x), where 4(x) > 0 for x X. The pomts x,, x2, x3 which satisfy x,
COMPETITIVE

Xl

X2

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x3

FIG. 2. An illustration of how capital intensity reversals can result in a point of period 3 for the policy function.

changes in the slope of the production frontier with changing capital) would have a maximum effect. Unfortunately, indifference curves cannot be specified independently of technology (i.e., independently of V) since future utility, W, depends on technology. This dependence decreases, however, as the discount factor decreases. In fact, since -Au/Ay = PA W(y), the nonlinearity of indifference curves decreases with 8. This leads to the conjecture that complicated dynamics of the policy function can occur when p is sufficiently small. Let

EXAMPLE.

V(,u, y) = xy - x2y - fy - .075y2 + yx

This

function

satisfies

our

standing

- 7x2 + 4x3 - 2x4.

assumptions

on

the

domain the

D= [0, 1.71 x [0, 1.71. It is also easily verified that, when fl=O.Ol,

three points x1 = sin2(n/7),

x2 = sin2(2n/7),

x3 = sin2(3n/7)

form a period three orbit (they satisfy the Euler equations that describe an interior solution). In fact, when b =O.Ol the policy function is given by h(x) = 4x( 1 -x) when 0 dx < 1. This can be verified using the fact that W(s) = yx - 5x2 is the value function for this problem.

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The policy function h(x) = 4x( 1 - x) (when 0 < x < 1) has an invariant measure which is ergodic and absolutely continuous with respect to the Lebesgue measure. The density of this measure is l/rc,/m. In addition, the policy function h(x) = 4x( 1 - x) has sensitive dependence on initial conditions: its Lyapounov exponent is log 2. A slight modification of the above example,

yields a problem (P) with policy function h(x, 8) = x( 1 -x)/(0.1 5 + lo/?), for discount factors in the interval (0, &I. Jacobson [lo] has proved that the one-parameter family of maps f(x, 2) = Ax( 1 - x) has absolutely continuous invariant measures for a set of parameter values of 1 having positive Lebesgue measure in ($ 43. Furthermore, it is known that for every II z 1, there exists a A, so that the mapf(x) = 1,x( 1 -x) has a stable periodic orbit of minimum period n, to which almost all trajectories are asymptotic. In fact, there is an open interval containing 1, for which this is the case (Collet and Eckmann [4]). In contrast to much of the earlier literature, we thus obtain policy functions with (almost) globally stable periodic orbits as well as policy functions with complicated dynamics and sensitive dependence on initial conditions. Furthermore, this set of examples is by no means exhaustive. In fact, Boldrin and Montrucchio [2] prove-in an n-dimensional setup-that any C”-function can arise as a policy function for some optimal growth model. Their result could also be derived, for the one capital good case, using the constructive approach implict in the 2x( 1 -x) examples. While the above examples rely on a one-period return function V satisfying all the requirements for a reduced form growth problem (A.l), (A.2), it is by no means clear at this stage that this particular V is in any sense “reasonable.” The answer is positive. We saw in Section 2 that every two-sector growth model yields a pair (V, D) satisfying (A.l), (A.2). It is possible to show the converse result, using a procedure similar to the one developed in Deneckere and Pelikan ([7], Proposition 6). Essentially, the method consists of showing that u can be taken to be linear and g to be Leontief, while keeping f concave and strictly monotone in each argument. It is also worth remarking on the “stability” of the example given above. The dependence of the policy function on V, /? and D is continuous in the Co-topology. That is, small changes in V, for instance, result in a policy fuction which is Co, but in general not C’ near the original policy function. These Co small changes in h may result, for example, in the creation of a stable periodic orbit where previously there had been an absolutely continuous invariant measure. Thus, the invariant measure (or measures) of

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the policy function does not depend continuously on the data of the problem (i.e., V, /I, and D). However, if h has, for example, a periodic orbit of period 3 which is hyperbolic (i.e., I(d/dx)j h3(x)j # l), this orbit will persist under C? small perturbations of the policy function, and the new policy function will certainly have complicated dynamics in the sense of Li and Yorke.

5. SUFFICIENT CONDITIONS FOR COMPLEX DYNAMICS We have seen that a necessary condition for complicated dynamics of an interior policy function is that V,,(x, y) take on different signs. The theorem below states sufficient conditions for complicated dynamics. It makes use of the following hypothesis: (A.3)

There exists a smooth function x = 4(y) so that

(i)

O<~(y)
(ii)

Vdx,y)>O

Moreover,

ifx<#(y),

the domain

V,z(x,y)
if x>d(y)

D is [0, 1] x [0, 11.

Proposition 5 implies that under hypothesis (A.3) h, is there exists 0 < a < 1 such that h, is nondecreasing on [O, a], on [a, 11, and its graph has no flat portion in the interior h& .) assumes its maximum at that point where the graph sects the curve x = +4(y). We can now state

unimodal (i.e., nonincreasing of D). In fact, of h,(x) inter-

PROPOSITION 6. Suppose that (A.3) is satisfied and that there exists an a in (0, 1) such that 0 = h,(O) = h,( 1) and h,(x,) = 1, where x1 = c#(1). Then there exists an open interval J containing a with the property that /?E J implies h&x) has a point of period three, and hence has complicated dynamics in the senseof L,i and Yorke.

Proof: The graph of h, covers the unit interval twice, since h,(x,) = 1 and h,(O) = h,( 1) = 0. This implies that h, has exactly one fixed point .U in the interval [x,, 11, because h, is nonincreasing on this interval (note that h, may have many fixed points on [0, x1]). By continuity of h,, and since h, is nonincreasing on (x1,1), there exist a in (0, xi) and b in (2, 1) such that h,(a) = h,(b) = x,. Again, since h, is nonincreasing on [b, 11 and maps that interval onto [0, xi], there is a point c in [b, 1) such that h,(c) =a. One has hd(b)=O, h:(c)= 1, so hi has a fixed point in (b, c). It cannot be a fixed point of h,, and thus is a periodic point of period 3. This property is preserved under small Co perturbations of h,. Since h, depends

continuously

on p (Proposition

1), the result follows,

1

24

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V2(L

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7. The hypotheses of Proposition

PROPOSITION minII

AND

O)l,

I VAO,

O)l}/V1(O,

O),

6 are satisfied with a = rovided that (A.3) holds and that P

I V2(x1, l)I/V,(l,O)~a<

1.

Proof: It is enough to establish that the path (x,, 1, 0, 0, O,...} is optimal from x,. Weitzmann [ 163 has shown that the following condition is sufftcient for the path {k:) to be optimal from k,*,

V(k,,k,+,)dV(k:,k,*,,)+PV,(k,*,,,k,*,,)(k;”+,-k,+,) - V,W:,k,*,,)(k:-k,)

(5)

for all {k,) such that (k,, k,,, ) ED and k, = k,*, provided that D is compact and V is C’ on D. The concavity of V on D implies V(k,,k,+,)dV(k:,k:+,)+

V,(k:,k:+,W-k:)

+ V,(k:,k,*,,)(k,+,-k,*,,).

(6)

By hypothesis, Vz(0, 0) + crV,(O, 0) < 0, v~(l,o)+~v,(o,o)6o, V&l

3 l)+aV,(l,O)bO.

Substituting these inequalities into (6), we see that Weitzmann’s conditions for optimality (5) are satisfied at /? = tl. 1

sufficient

The hypotheses of Proposition 7 are not vacuous: they are satisfied for the one-period return function V which led to the 4x( 1 - x) policy function in Section 4 (with a =O.Ol). An interesting issue that remains is what restrictions these hypotheses impose on the data (u, J g) of the two-sector growth model sketched in Section 1. While a complete analysis of this problem would lead us too far astray, we will answer the question under the assumption that u is linear and g is Leontief: g(x, I) = min(x/b, 1). First of all, (A.3) can be satisfied in this context. If 6 is small but positive, low levels of initial capital will be associated with little labor used in the investment sector. Thus, for low values of x, the consumption good sector will be more labor intensive than the investment sector (i.e., V,, > 0). For large values of x, however, the investment good sector will absorb a large fraction of the labor force, and the consumption good sector will be more capital intensive than the investment goods sector (i.e., V,, < 0). It can also be shown that the additional hypotheses of Proposition 7 imply that f must

COMPETITIVE

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25

exhibit a suflicient degree of decreasing returns to scale. In particular, constant returns to scale are ruled out under our maintained assumptions on u and g.

REFERENCES 1. J. BENHABIB AND K. NISHIMURA. Competitive equilibrium cycles, J. Econ. Theory 35 (1985), 284306. 2. M. BOLDRIN AND L. MONTRUCCHIO, On the indeterminacy of capital accumulation paths. University of Rochester, mimeo, 1985. 3. W. A. BROCK AND J. SCHEINKMAN, On the long-run behavior of a competitive firm, in “Equilibrium and Disequilibrium in Economic Theory” (G. Schwodiauer. Eds.), D. Reidel, Dordrecht, 1978. 4. P. COLLET AND J.-P. ECKMANN, “Iterated Maps on the Interval as Dynamical Systems,” Progress in Physics, Vol. I. Birkhauser, Boston, 1980. 5. W. A. COPPEL, The solution of equations by iteration, Proc. Cambridge Philos. Sot. 51 ( 1955), 4143. 6. W. D. DECHERT AND K. NISHIMIJRA, A complete characterization of optimal growth paths in an aggregate model with a non-concave production function, J. Econ. Theory 31 (1983), 332-354. 7. R. DENECKERE AND S. F’ELIKAN, Competitive Chaos, Institute of Mathematics and Its Applications, University of Minnesota, preprint No. 114, 1984. 8. J. M. GRANDMONT, On endogenous competitive business cycles, Econometrica 53 (1985). 99551046. 9. J. M. GRANDMONT, “Periodic and Aperiodic Behavior in Discrete One Dimensional Dynamical Systems,” CEPREMAP Working Paper, No. 8317, 1984. 10. M. V. JAKOBSON, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981). 39-58. 11. T. LI AND J. YORKE. Period three implies chaos, Amer. Mafh. Monlh1.v 82 (1975), 985-992. 12. L. MCKENZIE. “Optimal Economic Growth and Turnpike Theorems,” Discussion Paper, No. 79-1, Department of Economics, University of Rochester, 1980. 13. A. N. SARKOVSKII, Coesistence of cycles of a continuous map of a line into itself, Ukrain. Math. Z. 16 (1964), 61-71. 14. J. SCHEINKMAN, On optimal steady states of n-sector growth models when utility is discounted, J. Econ. Theory 12 (1976), 1 l-20. 15. W. SUTHERLAND, On optimal development in a multi-sectoral economy: The discounted case, Rev. Econ. Stud. 37 (1970). 585-589. 16. M. WEITZMANN, Duality theory for infinite horizon convex models, Management SC. 19 (1973) 783-789.