Journal of Economic Dynamicsand Control 4 (1982)37-55. North-Holland

A CHARACTERIZATION OVERLAPPING

OF ERRATIC DYNAMICS GENERATIONS MODEL

Jess BENHABIB

IN THE

and Richard H. DAY*

New York University,New York,NY 10012, USA The University of Southern California, Los Angeles, CA 90007, USA

ReceivedDecember1980 In this paper we characterizeand give examples of wide classesof utility functions which generateerratic dynamics in the standard, deterministic,overlapping generationsmodel. Erratic dynamics refers to feasible trajectories which are bounded but which do not converge to stationary points or periodic orbits. We show that such trajectories are Pareto-efficient.We introduce credit into the model and show how a constant credit expansion rate can result in erratic trajectoriesin prices and the real value of credit. Finally, we briefly discussce.rta$~ technicalaspectsof the mathematicsused and the ‘statistical’nature of the d@amics that can arisefrom deterministicdynamicalsystems.

1. Introduction The pure ‘consumption-loan’ model of Samuelson (1958) has been one of the most widely discussed subjects in the contemporary economic literature [see Meckling (1960), Cass and Yaari (1966, 1967), Diamond (1965), Thompson

(1967), Shell (1971), Starret (1972), Gale (1973), Cass, Okuno and

Zilcha (1979), Brock and Scheinkman (1980)]. Much of the discussion centered around the efficiency of the allocation (in the Pareto-sense) that results from overlapping generations. But even when efficiency is not at issue, the model has much to offer. In contrast to models which have a finite horizon (a built-in doomsday), the horizon in the consumption-loan model extends indefinitely. ‘Individuals, however, have a finite horizon and plan their lifetime consumption-saving path over their finite lifetime.’ As pointed out by Cass and Shell (1980), this allows the life of the economic system to extend indefinitely while not forcing the doomsday problem onto the agents in the economy or onto some representative individual. And in contrast to other infinite hkizon models, it does not require economic agents to have *The illuminating and helpful discussionswith ProfessorsW.A. Brock and D. Cass are happily acknowledged. ‘The sequentialgenerationsmodels of Leontief (1958),Day (1969) and Day and Fan (1976) also share this property. Their mathematical structure is similar to that of the overlapping generationsmodel.

0165-1889/82/0000-0000/$02.75

0 1982 North-Holland

38

J. Benhabib

and R.H.

Day,

Characterization

of erratic

dynamics

perfect foresight up to infinity. In the standard Samuelson version, the heterogeneity of consumers is restricted to the simultaneous existence of the young and the old. Recently, Cass, Okuno and Zilcha (1979) introduced heterogeneity within generations and obtained illuminating results on the existence and Pareto-efficiency of equilibrium trajectories. It is the possibility of oscillitary trajectories on which we focus our attention in the present paper. In particular we examine non-stationary paths where net trades between generations remain positive and feasible but exhibit an erratic character. We first characterize a class of utility functions, assumed to be unchanging over the generations, which give rise to such erratic trajectories. These trajectories essentially are paths that do not converge to periodic orbits or stationary points but remain bounded. Co-existing with such trajectories are also periodic orbits of arbitrarily large periods, which for all practical purposes are indistinguishable from the aperiodic ones. What is surprising is that such trajectories arise from a very wide class of utility functions that are robust with respect to perturbations in the parameters of the system (see section 6). In section 4 we discuss the specific mechanism by which intergenerational exchanges can take place [on this see also Gale (1973, sect. LC)]. This requires the introduction of credit (say in the form of checking accounts) which is extended to the agents by a central authority. The expansion of credit is controlled by this central authority via a nominal rate of interest. We show that the path of intergenerational trades are insensitive to this interest rate. A nominal rate fixed at a given level, for instance, results in a constant rate of expansion of the nominal value of credit. Yet, we show that the real value of credit can fluctuate erratically. What of the Pareto efficiency of intergenerational exchange in a chaotic regime? Using results of Balasko and Shell (1980) and Okuno and Zilcha (1980) it is possible to establish efficiency for all periodic orbits and erratic trajectories. This is done in section 5 of this paper. Our discussion concludes with a review of some of the technical aspects of chaotic dynamics and some possible extensions of our work. Of particular interest is the potential use of ergodic theory to obtain a statistical description of the deterministic trajectories of key variables (intergenerational trades, prices, the real value of credit) in the chaotic region. 2. Intergenerational

exchange

2.1. The overlapping generations model

Following Gale (1973) we consider exchange between a population of overlapping generations growing at a rate y. A representative individual lives for two periods, and, when young, determines a (non-negative) consumption for his youth c,,(t) and for his old age c,(t+ 1). His preferences are

J. Benhabib and R.H. Day, Characterization of erratic dynamics

39

represented by a utility function V(c,(t),c,(t+ l)), and he receives an endowment w. in his youth and wr in his old age. The interest factor at time t, P, (defining the exchange rate between present and future consumption) determines the representative individual’s budget constraint,

cl@+l)=w, +PrCwo-co(~n c&)20,

c,(t+ l)ZO, I

(1)

Since by assumption the aggregate endowment grows at the rate y, the market-clearing equilibrium condition for the economy as a whole is (l+y)[w,-c,(r)]+w,-c,(r)=O.

(2)

This materials balance constraint, together with the budget constraint (1) defines the set of feasible programs for this economy. Define (c;(t), cr (t + 1)) to be the consumption vector maximizing the utility of the tth generation subject to its budget constraint given by eq. (1). Let U(c,*(t), c:(t + 1); pI, wo, w,)= U*(p,, wo, wl). Then a dynamic, pure exchange equilibrium consists of price and consumption sequences in which the latter are individually optimal and consistent: Definition I. A pure exchange equilibrium vectors (p,, c,(t), q(t)),“, r such that, for all t,

trajectory

W,(t), cl@+ 1);A, wo,w,)= u*(P,, wo,WI),

is a sequence of

(3)

and the materials balance constraint (2) holds. Members of each generation may either save or borrow in their youth, thereby carrying claims or debts into their old age which they then settle with members of the new generation. They either claim their savings plus interest or pay their debts, which correspond, in the same order, to the new generation’s savings or borrowings. The equilibrium price or interest factor sequence assures that the claims (debts) of the old generation are equal to the savings (borrowings) of the young generation. The institutional elements to assure the settlement of claims can range from a social security system to a central clearing house dealing in I.O.U.‘s [see Gale (1973) and section 4 below]. Alternatively, the existence of a non-perishable and negotiable asset can also serve as a convenient medium of exchange and a store of value. Later on we will explore a model where we introduce ‘credit’ to serve as a medium of intergenerational exchange. Gale named the case where the young exhibit impatience and borrow from the old, ‘classical’, and the case where the young save and lend to the old, ‘Samuelson’. Which of the two cases obtains depends on the utility functions

J. Benhabib

40

and R.H.

Day,

Characterization

of erratic

dynamics

and the endowments. In either case, whether the ‘classical’ or the ‘Samuelson’ case is feasible, there exists a no-trade equilibrium associated with a constant price sequence: if in period zero there is no exchange, then there will be no exchange in future periods. The classical no-trade equilibrium is locally unstable: for any amount of borrowing by the young in period zero, no matter how small, the market will not return to the no-trade equilibrium. Conversely, Samuelson’s no-trade equilibrium is locally stable [see Gale (1973, theorem 4)], This has led Gale to conclude that while the ‘Samuelson’ case is logically consistent, the case relevant to the real world (and to the theory of interest) is the ‘classical’ one. The issue of how and why any trade should take place between the young and the old has been discussed by many authors. In period zero, why should the young lend to the old who will not be around to repay their debts in the next period? And why should the old lend to the young if they will not be around to collect. In the Samuelson case where the young are the lenders, they could be compensated when they become old by the forthcoming young generation, which in turn could be compensated when they become old, etc. That time never ends must be taken as axiomatic for infinitely farsighted individuals or they will not lend when young since, in an economy of finite duration, the ‘last’ young generation will have no compensation for their savings. In the classical case the old repay debts of their youth. But this leads to an infinite regress, possibly to exchange among the prehominids. Let us, therefore, consider an initial situation in which, for whatever reason (government taxes or transfers in the first period, altruism, small perturbations which destabilize the no-trade equilibrium) that trade takes place. 2.2. The dynamics of the classical case In order to characterize the dynamics of the pure exchange equilibrium trajectory the following assumptions are invoked: Assumption 1. The utility function for the representative generation is strictly concave, twice differentiable, increasing in its arguments and either separable or homothetic. Assumption the utility c,(t+l)>O appropriate

2. For the prices along the equilibrium path, the solution to maximization problem of each generation is interior, ‘i.e., co(t), for t=O,l,... . (This would hold if the utility function satisfied Inada conditions.)

Under Assumption 2 the first-order conditions for the utility maximization

.I. Benhabib

and R.H.

Day,

Characterization

41

of erratic dynamics

problem (3) reduces to

&=

Uo(co(O, Cl@+ 1)) U,(c,(t),c,(t + 1))’

in which U0 and U, are the partial derivatives of U. Substituting the budget constraint (1) we get UoMt),cl(t+

1))

U1(co(t),c,(t-k

m=

Wl

--cl@+ c,(t)--0

this into

1)

(5) ’

Our problem

now is to reduce (5) to a difference equation in co(t) [or difference equation we confine our attention to the classical case. (For the problems presented by the ‘Samuelson’ case see the next section.) Gale (1973, theorem 5) p&ides the prerequisite result: c,(t)]. To obtain a well-defined

Lemma I. In the classical case and given Assumptions I and 2 above, the function (5) can be solved uniquely for c,(t + 1). Call this function c,(t+

I)=

G(co@);

wo,

(6)

4.

Now define the constrained marginal rate of substitution

W,(t); wo,WI):=

(CMRS) function,

~OMO~Cl@+ 1)l ~,C%m&+N

(7)

where use is made of (6) to eliminate c,(t -t 1). The function V describes the marginal rate of substitution of present for future consumption for individually optimal and feasible programs, i.e., programs which satisfy the individuals’ budget constraints. Combining (7) with the equilibrium condition (2) we obtain the difference equation’

L W,(t); wo,w,)(c,(t)- wo):= f(co(t)). co(t+l)=-w,+l+y Once trade begins in the classical case This result is readily established from = U,/U, >O, for all c,,(t), c,(t + l)>O difference equation (8) characterizes trajectories when c,(O) > wO.

the trajectory must remain classical. (8) by observing the fact that pt and using induction. Hence, the the pure exchange equilibrium

‘Solve (2) for c(t+ 1). substitute into (1) and cancel to obtain c(t+ l)=@,/l Then (8) follows from (4) and (7).

+y)(w,,-cc,(t)).

42

J. Benhobib

and R.H.

Day,

Characterization

of erratic

dynamics

Gale (1973, p. 23) asserted that in the case of cycles the trajectory would converge either to the stationary state or to a limit cycle. However, we show that cyclical trajectories may oscillate without ever converging to a cycle of any order. Before establishing this result we will briefly consider the dynamics of the Samuelson case. 2.3. The Samuelson case

So far we have only treated the case where the young people borrow. This is because we have been able to unambiguously define the dynamics of exchange only for c,(t)> w. (see Lemma 1). This is riot simply a technical difficulty because cyclical behavior implies an ambiguity in the dynamical system when young people lend to the old. This is illustrated in fig. 1 [see also Cass, Okuno and Zilcha (197911. The line XY is the materials balance constraint, AF is the offer curve, A is the stationary no-trade equilibrium, and D is the stationary exchange equilibrium. Note that to have nonmonotonic trajectories, AF must cut XY at D with a positive slope. But this implies that for a given c,(t),c,(t + 1) and therefore co@+ 1) is not uniquely determined. Only if an additional selection criterion is imposed can we discuss the dynamics of exchange and prices. One possibility, for example, is random selection. On the other hand, we could assume that when multiple equilibria are possible, the one yielding the highest utility to the young obtains, at the expense of the yet unborn. Assume that the youth of each generation can obtain the equilibrium price yielding the highest utility and that the offer curve is as shown (with EF steep) in fig. 1. At some point along the equilibrium path the young want to lend AB in return for BF in their old age. But to the youth of the following generation such a sacrifice is not acceptable at any price. Therefore if we

Fig.

1

J. Benhabib

and R.H.

Day,

Characterization

of erratic

dynamics

43

insist on equilibrium, the feasible exchange for the present young is AB for BC, where BC is substantially less than BF. From such a point on, however, if equilibrium and feasibility is to hold, the system must converge to the notrade equilibrium point A.

3. Erratic exchange equilibria 3.1. The chaos theorem

By erratic exchange equilibria we mean trajectories (p,,c,(t), c,(t+ 1)); whose components do not converge to a stationary state or limit cycle but remain bounded. Under conditions sufficient to generate such behavior limit cycles of all orders will also exist. All this is made precise in a theorem proved by Li and Yorke in their paper ‘Period Three Implies Chaos’. We first define the following: Definition 2.

f’(x) =x.

The iterated map f”(x) is defined as fk(x)=f(fk-i(x)) where A point x is k-periodic under f if fk(x) =x and fi(x) # x for

Q
The basic theorem which we will apply to the consumption following [Li and Yorke (197511: Chaos Theorem. equation X

Let

J be an interval

loan model is the

in R and consider the difference

t+1=f(x,),

(9)

in which f is a continuous mapping of J+J. such that

Suppose there exists a point XE J

f3(XEX
(10)

Then:

(i) For every k= 1,2,3,. . ., there exists a k-periodic solution such that X,E J for all t.

(ii) There is an uncountable set (containing no periodic points) SE J such that for every x0 ES the solution path of (9) remains in S and (a) for all x, YES, X#Y,

lim sup 1[f’(x) - f’(y)] I> 0,

t-r03

lim inf 1f’(x) - f ‘Q I = 0;

t+m

44

J.

Benhabib and R.H. Day, Characterization of erratic dynamics

(b) for all period points x and all points YES, lim sup 1f’(x) -f’Q t-m

I> 0.

We note that statement (ii-a) of the theorem implies that trajectories in the set S starting from different points eventually get arbitrarily close but then diverge again. Definition 3. If f is a continuous mapping of an interval J into itself and there exists some initial x for which (10) is satisfied, then the difference equation (9) will be said to be chaotic on J. Trajectories in the chaotic set S of the chaos theorem will be said to be erratic. [For an illuminating discussion of chaotic difference equations, see Guckenheimer, Oster and Ipaktchi (1977).]

Letting z,=q,(t) and defining f as in eq. (8) we can apply the chaos theorem to characterize erratic exchange equilibria. The question now to be examined is: ‘Since the qualitative behavior of trajectories of (8) depend on the utility function V(*, a), what characteristics of the utility function are sufficient to generate chaos in the sense of Li and Yorke? 3.2. The sufficient substitutability

condition

We will characterize chaotic trajectories in terms of the constrained marginal rate of substitution function, V(a), defined in (7). Indeed, the sufficient condition about to be stated essentially requires the isoquants of the utility map to have enough curvature so that the marginal rate of substitution can vary sufficiently: SuJkient Substitutability

Condition.

The utility function is said to satisfy the condition if there exists a E > w. such that

sufficient substitutability

0) al= (l+y1 >v(e) > 1 (ii)

1 a2= l+y

(iii)

O
(resp. < 1)

( )

V(a,c^+(l -al)wo)> 1 -l+y

(

V(a,a,?+(l

1

1 (resp.2 1)

-ala,)wo)$

)

As before, y is the rate of population

growth.

0

J. Benhabib

and R.H.

Day,

Characterization

of erratic

dynamics

45

It is easily shown that the functions satisfying the ‘sufficient substitutability’ condition form a very large class. Note that the condition requires (i), (ii) and (iii) to hold for some 2, not all 15.Condition (i) simply requires that V(e)>(l +y). The function V(e) then defines an ai, which together with c^, we and y define a second point at which V(c) has to be evaluated. Condition (ii) puts a restriction on V(c) at that point; it also must be greater than (1 + y). Note that c0 > we implies that this new point is greater than e. Finally, condition (iii) uses the value of this second point to define a third point at which to evaluate V(c). This point falls to the right of the second point and (iii) requires V(c), evaluated at this third point, to be less than or equal to (1 + y)/a,a,. Fig. 2 illustrates the ‘sufficient substitutability’ condition.

Fig. 2

It is obvious from inspection that the construction of such functions is very easy. Essentially it requires V(c) to be steep at first and then to taper off sufficiently. [For example, let w. = 0, y = 0, c^= 1. Then choose V(c) such that 1/(1)=2, V(2) = 1.5 and Y(3)=0.3. Such a V(c) will satisfy the condition given above.] When the utility function is separable and linear in old age consumption, the constrained marginal rate of substitution function V(c,) is simply the marginal utility of youth consumption. That is, when U(co, cr) = u(co) +cr, we have V(c,)=u’(c,). For this special class of utility functions, the requirements of the ‘sufficient substitutability’ condition reduce to #(co) being sufficiently concave. Fig. 3 illustrates this.

46

J. Benhabib

and R.H.

Day,

Characterization

of erratic

dynamics

Fig. 3

3.3. Chaos in the consumption loan model

We are now ready to present our basic finding: Theorem function condition. wo + w,/u

1. Let Assumptions 1 and 2 hold and assume that the CMRS V(c,(t); wO,wl) of eq. (8) satisfies the sufficient substitutability Then the difference equation (8) is chaotic on the interval J=(w,; + YN-

Proof: First Furthermore,

we note that f(co(t); wo) as defined in (8) is continuous. from (5) and (7), (c,(t + 1) - wo)( l+ y) = { U,(c,(t), c,(t + 1)) /U,(c,(t), c,(t+ 1))] (c,(t)-wo)=wl -c,(t+ l)O by Assumption 2. Because we are in the classical case c,(O)~(w~; w. + w,/(l +y)) implies that c,(t)E(w,; w,+ w,/(l +y)) for all t. We will now show that there exists a co E J such that f3(co) SC, < f(co) < j2(co). Consider first part (i) of the sufficient substitutability condition given above. Multiplying both sides of the inequality by (co - wo) and adding w. to both sides, we obtain 1

co -c--V(c,)(c,

l+Y

- wo) -I- wo =f(c,).

Thus co
f(co)
J. Benhabib

and R.H.

Day,

Characterization

of erratic

dynamics

47

and add w,, to both sides. This yields 1 f(co)= -V(c,)(c,-w,)+we=a~c~+(1-a,)w, 1 +y 1 <(~1(co--wo)) -V(a,c,+(l-acr,)w,)+w, 1 +y

so that f(co)
_ 1 :,

RV(B)+wo=f(f2(Co))=f3(CO)7

-

where B=

&&(co-wo)‘(co)

Thus co 2_f3(co) and the conditions for the Chaos Theorem are fulfilled. This concludes the proof. IJ 3.4. Examples

(i)

Consider the concave utility function3

(11) where k=a + w. and a, A and k are positive constants and wo’ is the youth endowment. The marginal rate of substitution, e’(l-((Co-Wo)“), is always positive for any finite co. The equilibrium sequence for our model then becomes, for zero population growth, c,(t+

l)-

w.

=ea(’

~~~cO~f)~wO)lo)(~o(~)-~o),

‘The utility derived from lirst-period consumption functional form exhibiting constant risk aversion.

(12)

is described by the standard, widely-used

48

.I. Berikubib

and R.H.

Day,

Characterization

of erratic

dynamics

or x 1+1=e a(1- (-M.JJ x,=rx,emx’,

(13)

where r=e” and co(t)- w0 =x,. This equation satisfies the conditions of Theorem 1 for a>2.692 (or r> 14.765). For example, let r= 100 and x0 =O.Ol on eq. (7). Simple calculations show xJe”-‘, then c,(t + 1) - w. < w1 and c,(t + 1) > 0.4 (ii)

Now suppose the utility function for each generation is given by OScoSa/b,

U(c,(t),c,(t+1))=ac,-+bc;+c,,

a,b>O,

and let the endowment vector be (w,, wl)=(O, 5) with G>a/b and let the population be stationary. Then the difference equation describing the dynamics is given by c,(t + l)=ac,(t)(l

-(b/a)c,(t)).

(14)

Note that c,(t) E [0, a/b] for all c,(O) E [0, a/b], provided a 5 4. In fact, co(t + 1) attains a maximum for co(t)=a/2b. Eq. (14) can be shown to satisfy the requirements of our theorem for UE [3.53,4], b =a. For a = 3.83, for example, (14) has (approximately) a 3-period cycle for Zo=0.1561 and where F(c) =0.5096 and F*(~?~)=0.9579. [For further discussion, see Marotto (1978, example 4.1) or Hoppensteadt and Hyman (1977).] Note that for c,=a/b utility saturates. But the erratic trajectories and those with period greater than one never attain b/a since if c(t) = b/a, c(t + i) = 0 for all i = 1,2,. . . . (iii)

For a final example let the concave utility function be

WC,,Cl)= 4Since c,(t)--,, satisfied.

A(co+b)l-‘+c

l-a

1,

~20,

a#:,

A>O,

bz0.

(15)

and cl(t) are positive along the erratic paths, Assumptions 1 and 2 are clearly

J. Benhabib and R.H. Day, Characterization of erratic dynamics

49

.-

This leads to the difference equation c,(t+l)-we=-

A 1+ Y kl(d

1 - %I + 4

(co@)-WA

’

(16)

where k= b + w. and y is again the rate of population growth. For large values of A (2 50) and a (15) again chaos emerges [see Hassel, Lawton and May (1976) and May and Oster (1976, table l)].

4. Consumption loans with the social codivance

of ‘credit’

So far we have not discussed the mechanism by which trades between generations take place. It turns out that this is a delicate matter. In the ‘Samuelson’ case young people sell part of their endowment for a negotiable asset which they in turn use to purchase goods in their old age. This negotiable asset performs one of the functions of money: it serves as a store of value. In the ‘classical’ case the yougn borrow and the old settle their debts. This requires an intermediary or a central authority who extends credit to the young who wish to borrow, say in the form of checking accounts. The young then buy goods from the old with checks; who extends credit to the young (mortgage loans, for instance) who wish to borrow, say in the form of checking accounts. The young then buy goods (for example, houses) from the old with checks; the old in turn deposit these checks and settle their debts. The checking accounts demanded by the young and provided by the central authority as credit then serve as a medium of exchange between generations.5 The central authority can also try to regulate the amount of credit by stipulating a nominal interest on credit. We will see that erratic equilibrium trajectories in the real value of credit may arise even though nominal credit expands at a constant rate. The representative young consumer maximizes his two-period utility function, U(c,(t), c,(t+ l)), subject to

P(OCo(~) =P(GWo +w, p(t+ l)c,(t+

l)=p(t+

(17)

l)w, -(l +a)m(t),

(18)

where m(t) is the balance of the checking account that he obtains as credit and D is the interest on m(t) that he has to pay in the subsequent period ‘In general the young may on balance either borrow from or lend to the old and a more complete analysis would require a model where generations live longer than two periods. For a formal treatment of this, see Gale (1973, sect. 6).

50

J. Benhabib

and R.H.

Day,

Characterization

of erratic

dynamics

when he is old. The materials balance constraint, as before, is (19

(1+y)(w,-cc,(t))+w,-c,(t)=O. Substituting for c,(t + 1) from (19) (updated substituting for m(t) from (17) into (18) we obtain c&+1)-w,=+$

a period)

--&co(t)-WJ.

p(t)=( ) 1

l+c

(18) and

w-4

The first-order conditions for the agent’s utility maximization interiority as in the previous section)

PO + 1)

into

yield (assuming

U&o(~), cl@+ 1)) ~,Mwl@+1r

(21)

and (20) becomes

( > 1 1 +Y

c&+1)-ww,=

~&O(~)> Cl@ + 1)) (c,(t) _ wo). ~,Mo,c,(~+ 1))

(22)

Since l/(1 +y)>O, the results of Lemma 1 in the previous section still apply and we have v(c

@). w. w Or 91

)=

~&cJ(~)~ Ul(co(o9c,(~+

cl@+

1)) 1))’

(23)

Eq. (22) can now be written as a well-defined difference equation,

co(t+l)=l+y -L %o(O)(co(~) - wo)+ wo. Since eq. (24) is the same as eq. (8), the results of Theorem 1 apply as before. Note that fixing (T determines the rate of expansion or contraction of m(t), the level of credit. Along the equilibrium trajectory, the young who obtain m(t) will have to acquire (1 +a)m(t) in the subsequent period to repay their debts. They can obtain this when they are old by selling part of endowment to the young who have credits equal to m(t+ l), which they obtained from the central authority. Thus in equilibrium, we must have m(t+ l)=(l

+a)m(t).

(25)

51

J. Benhabib and R.N. Day, Characterization of erratic dynamics

A fixed Q implies a fixed rate of credit expansion. In spite of this, the trajectory of real debt can be quite erratic. This is immediately seen from (17) which can be written as

mW(t) = co(t)- wo. If intergenerational

(26)

trades are erratic, so is real debt.

5. Efficiency properties of erratic and chaotic paths Using recent results of Balasko and Shell (1980) and Okuno and Zilcha (1980), it is possible to show that the periodic and erratic paths described in Theorem 1 are Pareto-efficient. To use the Balasko-Shell or Okuno-Zilcha results we have to assume the following: Assumption 3. The Gaussian curvature consumption bundles along the equilibrium away from zero.

of the indifference curves at trajectory is uniformly bounded

This assumption rules out the possibility that the indifference curves are close to being flat in the neighborhoods of the consumption bundles along the equilibrium trajectory. 6 We can now prove the following efficiency theorem: Theorem 2. Let Assumptions 1, 2, and 3 hold. Then all the periodic and erratic trajectories other than the no-trade equilibrium, described by Theorem 1 are Pareto-efficient. Proof We will use Theorem 3A of Okuno and Zilcha (1980), which states that provided Assumption 3 is satisfied, the equilibrium trajectory for the overlapping generations model is efficient if CE i l/l 1p(t) 11= co. This will be true if the sequence { l/l1 p(t) II}; contains an infinite subsequence which is bounded away from zero, that is if

lim sup l/l I p(t) ( I > 0.

1-w

(27)

From eq. (20) or (4) we have l/PO+ I)= wo(w/Pw

(28)

“For instance, the utility ffkction in example (i) of section 3.4 would satisfy this assumption.

52

J. Benhabib and R.H. Day, Characterization of erratic dynamics

Without loss of generality we can normalize the price sequence so that l/p(O) = c,(O) - wO. For a stationary population,’ eq. (7) (or (22)) becomes

coo+ 1)- wo= W,(t) - wo)+woNo(d - wo),

(2%

or x(t + 1) = V(x(t) +

w,,x(t),

(30)

where x(t)=c,(t)-wo. Thus, the trajectory {l/p(t)}: is the same as {x(t)}?. The no-trade equilibrium, x(t)=0 [that is c,(t)- w. =O] is a periodic point of period one. Thus by (ii-b) of the Chaos Theorem in section 3.1, lim,, m sup 1x(t)-0 ( > 0 for any erratic path. This is also obviously true for any periodic orbit with x(O)#O as well. Since l/p(t) =c,(t) - w. for all t as shown above, the theorem is proved. 0 6. Final remarks We conclude by briefly pointing out certain technical aspects of erratic or chaotic dynamics since this is a relatively new subject: (1) Erratic (or chaotic) trajectories seem to be a ubiquitous feature of nonlinear difference equations. That such dynamics can be robust (structurally stable) with respect to small perturbations of the map defining the difference equation has been shown by Butler and Pianigiani (1978) and by way of example by Smale and Williams (1976). See also Kloeden (1976). (2) A characterization of chaotic behavior in higher order systems has been given by Phil Diamond (1976) as well as Marotto (1978). (3) Chaotic trajectories arise in differential equations as well. A number of studies have exhibited such possibilities (called ‘strange attractors’) in differential equation systems of order higher than three [see Lorentz (1963) and Ruelle and Tackens (1971)]. ‘If population is growing at the rate y, then p(t) must be interpreted as present value prices which have been further discounted by the rate of population growth, as Okuno and Zilcha (1980) pointed out. Thus p(t)=n(t)/(l +y)‘, where n(t) is the price at t, not discounted by population growth. However, eq. (29) then becomes c.(t+l)-w,=il;;V((c,(r)-w,)C*,)(c,(t)-W,),

(29’)

and (27) becomes Mt + I)= wl(tN(v4t)). (27’) Normalizing such prices that n(0) = c,(O) - w0 and combining (27’) and (29’) we obtain l/lr(t)=(l/(l+y)‘)(c,(t)-ww,) or l/p(t)=(l+y)lln(t)=c,(t)-w,. Since the theorem of Okuno and Zilcha is given in terms of l/p(t), the proof of Theorem 2 above applies to the case of population growth as well.

J. Benhabib

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(4) The dynamics within erratic sets may often be indistinguishable from stochastic processes, sometimes even from simple Bernoulli processes [see Guckenheimer, Oster and Ipaktchi (1977) or May and Oster (1976)]. This has led to the use of statistical techniques and ergodic theory to investigate the average behavior of erratic trajectories. One can denote the fraction of that are in an interval [~,,a,] (or take a iterates {x, F(x), . . ., F”-‘(x)} Markov partition of the set to be studied). The fraction can be represented by cp(x, n, [a,, aJ) and one can take the limit as n+o3 if such a limit exists. The subject of ergodic theory is the existence of limiting fractions for intervals [a,,~,]. Such limiting fractions are of particular interest if they are independent of the initial point x. For the model discussed in the previous section such results would allow us to characterize a limiting long-run average of net trades and real balances even though such quantities fluctuate erratically from period to period. For a given initial value of net trades it is sometimes possible to show that the limiting function cp(x,N(a,,a,)) exists. From Theorem 1 we know that the function defining net trades I;(@); wO) maps the interval J= [w,; wJ(1 +r)] into itself. We can then apply the Kriloff-Bougolioubof Theorem’ to deduce the existence of an invariant measure on J. Thefi the Birkhoff Ergodic Theorem (1936) [see also Halmos (1956)] asserts the existence of the limiting fractions for ‘all initial x E J except for a subset of measure zero’. The trouble is that the invariant measure defined on J, since it is not continuous, may be supported on a very small set or even on a point. In such a case ‘all initial x EJ except for a subset of measure zero’ may be just one point. When erratic paths (as defined in Theorem 1) exist, however, it is possible to show the existence of continuous invariant measures (assigning zero measure to points) on J [see Lasota and Yorke (1977) and Pianigiani (1979a)]. Then the set of initial values for which time averages exist cannot be supported on isolated points. In certain cases the limiting fractions can be independent of the initial value.g In example (ii) of section 3.4, for the value of ‘u’=4, b = 1 [see Stein and Ulam (1974), or for ‘u’=3.6785735 . . . see Ruelle (1977)], such independence can be obtained. In fact for this difference equation it can be shown that there are an infinite number of such values of ‘a’ [see Pianigiani (1979b)]. 8For an exposition, see Pianigiani (1979a). ‘BirkhoBs Erogodic Theorem shows what is needed for independence from initial values (i.e., ergodicity) is ‘metrical transitivity’. ‘Metrical transitivity’ requires that under the transformation considered there do not exist two disjoint invariant sets of positive measure. In practice,t&s is extremely hard to check for.

References Balasko, Y. and K. Shell, 1980, The overlapping-generations model, I: The case of pure exchange without money, Journal of Economic Theory 23,281-306.

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