Bhattacharya R., Majumdar M. Random Dynamical Systems: Theory and Applications

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1.9 Some Applications 103

Substituting Equation (9.69) in Equation (9.71) we get

t+1

= T ( p

, e,λ) = p

+ λ[2β/p

− 4(1 −α)]. (9.72)

Equation (9.72) is the law of motion of dynamical system parameter-

ized by e = (α, β) and λ ∈

. We then have

Proposition 9.10 There exist open sets N ⊂ E

and V ⊂

such

that if e = (α, β) ∈ N and λ ∈ V then the process (9.72) exhibits Li–

Yorke chaos.

Proof. Let K ⊂ E

and  ⊂ R

be compact sets containing

e =

(3/4, 1/2) and

λ = 361/100 in their respective interiors. For e =

e and

λ =

λ, Equation (9.72) becomes

t+1

= T ( p

λ) = p

+ 361/(100 p

) − 361/100. (9.73)

For convenience, we shall write T ( p,

λ) simply as T ( p). Let S de-

note the interval [0.18, 17]. If p

∈ S then p

t+1

∈ S as well. This follows

because the global minimum of T is at 1.9 (from the ﬁrst and second-

order conditions) where T takes the value 0.19 > 0.18. Likewise the

maximum value of T is at 0.18 where it takes the value 16.63 < 17.

Hence T maps the interval S strictly into the interior of S.

It is then easy to verify that T satisﬁes condition (3.1) on the inter-

val S, i.e., that there are points a, b, c, d in S such that c < b < a < d

and T (a) = b , T (b) = c, T (c) = d. Speciﬁcally, let a = 3.75. Then

b = T (a) = 1.9, c = T (b) = 0.19 and d = T (c) = 15.58 > a. Thus

the process (9.73) exhibits Li–Yorke chaos on S.

Clearly, the conditions of Proposition 3.1 are satisﬁed by the map T

(regarded as a function from S to S). Hence, there exists some ε>0

such that if g ∈ C(S) satisﬁes

sup

p∈S

|T ( p) − g( p)| <ε, (9.74)

then g also exhibits Li–Yorke chaos on S.

Recall that T ( p) ≡ T (p,

λ), where

e lies in the interior of the com-

pact set K ⊂ E

and

λ is in the interior of the compact set . Now, for

e = (α, β) ∈ K and λ ∈  the map T ( p, e,λ) given by Equation (9.72)

is clearly jointly continuous on the compact set S × K × . Let e



104 Dynamical Systems

another economy in K and λ



∈ . By uniform continuity, there exists

δ>0 such that if the (four-dimensional) Euclidean distance

d((p, e,λ), ( p



, e



,λ



)) <δ (9.75)

then

|T ( p, e,λ) − T ( p



, e



,λ



)| <ε. (9.76)

Let N × V be an open set in the interior of K ×  containing (

λ) and

having radius less than δ.Ifε ∈ N and λ ∈ V then for each p ∈ S,

d((p,

λ), ( p, e,λ)) <δ

and, hence,

|T ( p,

λ) − T ( p, e,λ)| <ε. (9.77)

Taking the supremum over all p ∈ S we note that (e,λ) ∈ N × V im-

plies T (·, e,λ) is within ε in the sup norm distance of T (·,

λ). Since re-

lation (9.77) is satisﬁed by every mapping T (·, e,λ) with (e,λ) ∈ N × V ,

the tatonnement process exhibits Li–Yorke chaos on an open set of

Cobb–Douglas economies, for an open set of “rate of adjustment”

parameters λ.

1.10 Complements and Details

Section 1.1. Simple rules of compounding interest and population

growth are early examples of dynamic analysis in economics. Professor

Samuelson in his Foundations talked about “a time when pure economic

theory has undergone a revolution of thought – from statical to dynamical

models. While many earlier foreshadowings can be found in the litera-

ture, we may date this upheaval from the publication of Ragnar Frisch’s

Cassel Volume essay.” Our quote at the beginning of Chapter 3 is from

this Frisch’s (1933) article. Frisch, Goodwin, Harrod, Hicks, Kalecki,

Lundberg, Metzler, Samuelson, and others made deﬁnitive contributions

to studying endogenous (short run) cycles in an economy.

A useful compendium of formal trade cycle models of Hicks, Samuel-

son , Goodwin, and Kalecki was already available in Allen (1956). Some

of the pioneering papers were collected in Gordon and Klein (1965) and

Cass and McKenzie (1974). We should stress that the developments went

signiﬁcantly beyond the scope of our presentation in this chapter in many

1.10 Complements and Details 105

respects (in particular, in the treatment of lagged variables). We should

also recall that difference, differential, and “mixed” systems (see Frisch

and Holme 1935) were all experimented with.

The models of Samuelson (1939) and Metzler (1941) are cele-

brated examples of “second-order” systems. In the Samuelson model,

one has

= C

+ I

, (C1.1)

(where Y, C, I denote income, consumption, and investment in period t).

= cY

t−1

, 0 < c < 1 (C1.2)

(where c is the fraction of I

t−1

consumed in period t).

= B[Y

t−1

− Y

t−2

], (C1.3)

where B > 0 is the “accelerator coefﬁcient.”

Upon substitution we get

= (C + B)Y

t−1

− BY

t−2

. (C1.4)

Subsequent research in dynamic economics continued in several di-

rections: descriptive models of growth and market adjustments, models

of intertemporal optimization with the duals interpreted as competitive

equilibria, models of intertemporal equilibria with overlapping genera-

tions of utility maximizing agents, and dynamic game theoretic models

capturing conﬂicts of interests. The emergence of chaos in simple frame-

works was noted by Day (1982, 1994) and Benhabib and Day (1981,

1982). Goodwin, with his life-long commitment to exploring nonlinear-

ity in economics (Goodwin 1990), remarked that if one were looking

for “a system capable of endogenous, irregular, wave-like growth,” the

“discovery and elaboration of chaotic attractors seemed to me to provide

the kind of conceptualization that we economists need.”

By now, we have a relative abundance of examples (that are variations

of the standard models) which generate complex or chaotic behavior.

The striking feature that has been duly emphasized is that complexities

arise out of the essential nonlinearity of the models even in their very

simple formulations. Moreover, the examples do not arise out of any

“knife-edge matching” of the parameters, making them atypical or acci-

dental. For useful review articles, see Baumol and Benhabib (1989), Day

and Pianigiani (1991), Grandmont (1986), Zarnowitz (1985). Some col-

lections of inﬂuential articles are Benhabib (1992), Grandmont (1987),

106 Dynamical Systems

Majumdar et al. (2000). The book by Day (1994) is an excellent intro-

duction to chaotic dynamics for students in economics.

Section 1.2. For additional material, see Brauer and Castillo-Chavez

(2001), Kaplan and Glass (1995), Day (1994), Devaney (1986).

Section 1.3. See Collet and Eckmann (1980), Devaney (1986), Majum-

dar et al. (2000, Chapter 13 by Mitra). Proposition 3.1 is contained in

Bala and Majumdar (1992) who dealt with the question of robustness

of Li–Yorke chaos in a discrete time version of the Walras–Samuelson

price adjustment process.

Theorem 3.1 (Proof of [2]) Let M be the set of sequences M =

}

∞

n=1

of intervals with

[A.1] M

= KorM

⊂ L, and α(M

) ⊃ M

n+1

If M

= K , then

[A.2] n is the square of an integer and M

n+1

, M

n+2

⊂ L.

Of course if n is the square of an integer, then n + 1 and n + 2 are

not, so the last requirement in [A.2] is redundant. For M ∈ M let

P(M, n) denote the number of i ’s i n {1,...,n} for which M

= K .

For each r ∈ (3/4, 1) choose M

={M

}

∞

n=1

to be a sequence in M

such that

[A.3] lim

n→∞

P(M

, n

)/n = r.

Let M

={M

: r ∈ (3/4, 1)}⊂M. Then M

is uncountable

since M

= M

for r

= r

. For each M

∈ M

, by Step 2, there

exists a point x

, with α

) ∈ M

for all n.

Let T ={x

: r ∈ (3/4, 1)}. Then T is also uncountable. For

x ∈ T , let P(x, n) denote the number of i’s i n {1,...,n} for

which α

(x) ∈ K . We can never have α

) = b, because then x

would eventually have period 3, contrary to [A.2]. Consequently

P(x

, n) = P(M

, n) for all n, and so

ρ(x

) = lim

n→∞

P(M

, n

) = r

for all r. We claim that

[A.4] for p, q ∈ T , with p = q, there exist ﬁnitely many n’s such that

(p) ∈ K and α

(q) ∈ L or vice versa.

1.10 Complements and Details 107

We may assume ρ( p) >ρ(q). Then P( p, n) − P(q, n) → x, and so

there must be inﬁnitely many n’s such that α

(p) ∈ K and α

(q) ∈ L.

Since α

(b) = d ≤ a and α

is continuous, there exists δ>0 such that

(x) < (b + d)/2 for all x ∈ [b − δ, b] ⊂ K. If p ∈ T and α

(p) ∈ K ,

then [A.2] implies α

n+1

(p) ∈ L and α

n+2

(p) ∈ L. Therefore α

(p) <

b − δ.Ifα

(q) ∈ L, then α

(q)  b,so

|α

(p) − α

(q)| >δ.

By claim [A.4], for any p, q ∈ T , p = q, it follows that

lim sup

n→∞

|α

(p) − α

(q)|≥δ>0.

Hence (3.2) is proved. This technique may be similarly used to prove that

(2(ii)) is satisﬁed.

Proof of (3.3) Since α(b) = c, α(c) = d, d ≤ a, we may choose inter-

vals [b

, c

], n = 0, 1, 2,...,such that

(a) [b, c] = [b

, c

] ⊃ [b

, c

] ⊃···⊃[b

, c

] ⊃···,

(b) α(x) ∈ (b

, c

) for all x ∈ (b

n+1

, c

n+1

) = c

, α(c

n+1

) = b

Let A =



∞

n=0

, c

], b

∗

= inf A and c

∗

= sup A; then α(b

∗

) = c

∗

and α(c

∗

) = b

∗

, because of (c).

In order to prove (3.3) we must be more speciﬁc in our choice of

the sequences M

. In addition to our previous requirements on M ∈

M, we assume that if M

= K for both k = n

and (n + 1)

, then

= [b

2n−(2 j −1)

, b

∗

] for k = n

+ (2 j − 1), M

= [c

∗

, c

2n−2 j

] for k =

+ 2 j where j = 1,...,n. For the remaining k’s that are not squares

of integers, we assume M

= L.

It is easy to check that these requirements are consistent with [A.1] and

[A.2], and that we can still choose M

so as to satisfy [A.3]. From the fact

that ρ(x) may be thought of as the limit of the fraction of n’s for which

(x) ∈ K , it follows that for any r

∗

, r ∈ (3/4, 1) there exist inﬁnitely

many n such that M

= M

∗

= K for both k = n

and (n + 1)

. To show

(3.3), let x

∈ S and x

∗

∈ S. Since b

→ b

∗

, c

→ c

∗

as n →∞, for

any ε>0 there exists N with |b

− b

∗

| <ε/2, |c

− c

∗

| <ε/2 for all

n > N . Then, for any n with n > N and M

= M

∗

= K for both k = n

and (n + 1)

,wehave

) ∈ M

= [b

2n−1

, b

∗

]

108 Dynamical Systems

with k = n

+ 1 and α

) and α

∗

) both belong to [b

2n−1

, b

∗

Therefore, |α

) − α

∗

)|  ε. Since there are inﬁnitely many

n with this property, lim inf

n→∞

|α

) − α

∗

)|=0.

Section 1.4. See Allen (1956), Baumol (1970), Devaney (1986, Chap-

ter 2), Goldberg (1958), Sandefur (1990), Elaydi (2000). Parts (a)–(c)

of Example 4.1 are contained in Majumdar (1974, Theorem 4.1). For an

elaboration of the survival problem, see Majumdar and Radner (1992).

Consider the difference equation

t+1

= Ax

, (C1.5)

where A is an n × n matrix and x

is a column vector in R

.Givenx

iteration of (C1.5) gives us

= A

. (C1.6)

The long-run behavior of x

is clearly linked to the behavior of A

Here is a deﬁnitive result:

Proposition C1.1 Let A be an n × n complex matrix. The sequence

A,...,A

,...of its powers converges if and only if

(1) each characteristic root r of A satisﬁes |r| < 1 or r = 1;

(2) when the second case occurs the order of multiplicity of the root 1

equals the dimension of the eigenvector space associated with that root.

For a proof see Debreu and Herstein (1953, Theorem 5).

Section 1.5. For possible generalizations of Proposition 5.1 one is led to

ﬁxed point theorems in partially ordered sets. See the review in Dugundji

and Granas (1982, Chapter 1.4). A detailed analysis of the survival prob-

lem treated in Example 5.2 is in Majumdar and Radner (1992).

Section 1.6. The monographs by Clark (1976), Dasgupta and Heal

(1979), and Dasgupta (1982) deal with the optimal management of ex-

haustible and renewable resources. For extensions of the basic qualitative

results summarized in Example 6.1, see a Majumdar and Nermuth (1982),

Dechert and Nishimura (1983), Mitra and Ray (1984), Kamihigashi and

Roy (2005), and the review by Majumdar (2005).

1.10 Complements and Details 109

Section 1.7. Consider S = [a, b] ⊂ R. Let S be the Borel sigmaﬁeld of

S, and P a probability measure on S.Ifα : S → S is measurable, then P

is invariant under α if P(E) = P(α

−1

(E)] for all E in S. P is ergodic if

“E ∈ S, α

−1

(E) = E” implies “P(E) = 0 or 1.” The dynamical system

(S,α) is said to exhibit ergodic chaos if there is an ergodic, invariant

measure ν that is absolutely continuous with respect to the Lebesgue

measure m on S (i.e., if S ∈ S, and m(E) = 0, then ν(E) = 0). In this

case, for brevity, call β an ergodic measure of α.

There are alternative sufﬁcient conditions for ensuring that a dynam-

ical system exhibits ergodic chaos. We (due to Misiurewicz 1981) recall

the one that is most readily applicable to the quadratic family.

Proposition C7.1 Let S = [0, 1] and J = [3, 4] given some

θ in J , de-

ﬁne α

(x) =

θ x(1 − x) for x ∈ S. Suppose that there is k ≥ 2 such that

y = α

(0, 5) satisﬁes α

(y) = y and |a



(y)| > 1. Then (S,α

) exhibits

ergodic chaos.

As an example, consider S = [0, 1] and

θ = 4. Here, α

(0.5) = 0,

(0) = 0 and α



(0) = 4 > 1, so the above proposition applies directly.

For this example, the density ρ of the ergodic measure ν has been calcu-

lated to be

ρ(x) =

π[x(1 − x)]

1/2

for 0 < x < 1.

The outstanding result due to Jakobson can now be stated:

Theorem C7.1 Let S = [0, 1], J = [3, 4] and α

(x) = θ x(1 − x) for

(x, 0) ∈ S × J . Then the set  ={θ ∈ J :(S,α

) exhibits ergodic chaos}

has positive Lebesgue measure.

In fact, more is known. Consider the interval J

= [4 − η, 4] for

0 <η≤ 1. Denote by m

the Lebesgue measure of the set of θ in J

for which the dynamical system (S,α

) exhibits ergodic chaos (for-

mally, m

= m(J

∩ )). Then, given ε>0, there is 0 <η<1 such

that

> 1 − ε.

110 Dynamical Systems

Thus, as η gets smaller (so that we are looking at θ “close to” 4), most

of the values of θ exhibit ergodic chaos (see Day (1994) and Day and

Pianigiani (1991) for more details).

Section 1.8. For a more complete discussion of bifurcation theory see

Devaney (1986, Section 1.12). By now there are many examples of

macroeconomic models with “endogenous” periodic behavior and bi-

furcations (in discrete and continuous time. See Azariadis (1993) and

Grandmont (1985) for very accessible presentations and references. Also

of interest are Torre (1977), Benhabib and Nishimura (1979), Becker and

Foias (1994), and Sorger (1994)).

Section 1.9.1. Following Chakravarty (1959) we outline the two-sector

Feldman-Mahalanobis model which was also prominent in the literature

on national planning.

C.1.9.1. The Feldman–Mahalanobis model portrays an economy (with

unlimited supply of labor) in which there are two sectors: the investment

good sector and the consumption good sector. Total investment in any

period t can be broken up into two components, one part going to augment

productive capacity in the investment good sector, the other part to the

consumption good sector.

Thus, investment I

is divided into two parts λ

and λ

where

> 0 and λ

> 0 are the fractions of investment allocated to the two

sectors. It follows, of course, that λ

+ λ

≡ 1. If β

and β

are the

respective output–capital ratios of the two sectors, it follows that

− I

t−1

= β

t−1

(C1.7)

− C

t−1

= β

t−1

. (C1.8)

Equation (C1.7) can be solved to get

= I

(1 + λ

)

, (C1.9)

where I

is the initial investment (i.e., investment in period t = 0); hence,

− I

= I

{(1 + λ

)

− 1}. (C1.10)

From Equation (C1.8) it can be deduced that

− C

= β

[{(1 + λ

)

− 1}/λ

]. (C1.11)

1.10 Complements and Details 111

Adding (C1.8) and (C1.9) we get

− Y

= I

{(1 + β

)

− 1}([β

/β

] + 1). (C1.12)

We can replace I

in (C1.12) by α

, where α

is the fraction of income

that is assigned to investment initially. We then have

= Y

[1 + α



+ β



{(1 + β

)

− 1}]. (C1.13)

What happens to the growth of income if we choose a higher value of λ

Note that (since β

, β

are technologically given) a higher value of λ

in-

creases the terms (1 + β

)

and lowers the term (β

+ β

)/β

While for small values of t the overall impact is unclear, the term

(1 + β

)

will dominate in the long run and will imply a higher Y

in future.

Section 1.9.3. For detailed expositions of the theory and applica-

tions of linear and nonlinear multiplicative processes, see Debreu and

Herstein (1953), Gantmacher (1960), Gale (1960), Morishima (1964),

and Nikaido (1968).

Section 1.9.4.1. The question of existence of a program that is opti-

mal (according to the Ramsey–Weizsacker criterion) in the undiscounted

model is a subtle one. Gale (1967), and McFadden (1967) treated this

problem in discrete time, multisector frameworks. A complete outline of

the existence proof for the one-good model in which the technology is

allowed to have an initial phase of increasing returns is in Majumdar and

Mitra (1982).

The problem of attaining an optimal program in a decentralized, price-

guided economy with no terminal date has also challenged economic

theorists. A particular problem is to replace the “insigniﬁcant future”

(IF) condition in Theorems 9.2 and 9.3 by a sequence of veriﬁcations

each of which can be achieved through the observation of a ﬁnite number

of prices and quantities. A symposium in Journal of Economic Theory

(volume 45, number 2, August 1988) explored this theme in depth: see the

introduction to this symposium by Majumdar (1988) and the collection

edited by Majumdar (1992).

On comparative dynamic problems, the reader is referred to Brock

(1971), Nermuth (1978), Mitra (1983), Becker (1985) Dutta (1991),

112 Dynamical Systems

Dutta, Majumdar, and Sundaram (1994) and Mitra (Chapter 2 in

Majumdar et al. (2000)).

Exercise 9.5 was sketched out by Santanu Roy (who should be respon-

sible for the blemishes!). In this context see Brock and Gale (1969), Jones

and Manuelli (1990), Rebelo (1991), K. Mitra (1998) and T. Mitra (1998).

Section 1.9.5. In his evaluation of the progress of research on optimal

intertemporal allocation of resources, Koopmans (1967, p. 2) observed:

In all of the models considered it is assumed that the objective of economic

growth depends exclusively on the path of consumption as foreseen for the

future. That is, the capital stock is not regarded as an end in itself, or as a

means to other end other than consumption. We have already taken a step away

from reality by making this assumption. A large and ﬂexible capital stock has

considerable importance for what is somewhat inadequately called “defense.”

The capital stock also helps to meet the cost of retaining all aspects of national

sovereignty and power in a highly interdependent world.

See Kurz (1968), Arrow and Kurz (1970), and Roskamp (1979) for mod-

els with “wealth effects.” In the literature on renewable resources and

environmental economics, incorporating the direct “stock effects” is ac-

knowledged to be a signiﬁcant ingredient in formulating appropriate

policies. Dasgupta (1982, p. 1070) summarized the importance of such

effects as follows:

As a ﬂow DDT is useful in agriculture as an input; as a stock it is hazardous for

health. Likewise, ﬁsheries and aquifers are useful not only for the harvest they

provide: as a stock they are directly useful, since harvesting and extraction costs

are low if stocks are large. Likewise, forests are beneﬁcial not only for the ﬂow

of timber they can supply: as a stock they prevent soil erosion and maintain a

genetic pool.

Benhabib and Nishimura (1985) identiﬁed sufﬁcient conditions for the

existence of periodic optimal programs in two-sector models.

Section 1.9.6. This exposition is based entirely on parts of Mitra (2000).

Boldrin and Montrucchio (1986) and Deneckere and Pelikan (1986) used

the reduced form (two-sector) model to provide examples of chaotic be-

havior generated by optimization problems. Of particular importance

also is the paper by Nishimura, Sorger and Yano (1994) on ergodic

chaos. Radner (1966) used a dynamic programming approach to pro-

vide an example of a multisector model (a linear-logarithmic economy)