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

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1.5 Increasing Laws of Motion 23

Now

α(

x) = α(t

x + (1 − t)0)

> tα(

x) + (1 − t)α(0)

≥ tα(

α(

tα(

α(

Hence

α(x)

is decreasing. Write B(x) ≡ α(x) − x. By [E.1]–[E.3]

there is some

x > 0 such that α



(x) > 1 for all x ∈ (0,

x]. Hence,



(x) > 0 for all x ∈ (0,

x]. By the mean value theorem

α(

x) = α(0) +

xα



(z), 0 < z <

≥

xα



(z)

α(

≥ α



(z) >α



(

x) > 1.

Now, if α is bounded, i.e., if there is some N > 0 such that α(x) ≤ N,

then [α(x)/x] ≤ [N /x]. Hence there is

x, sufﬁciently large, such that

α(

x)/

x ≤ [N /

x] < 1.

If α is not bounded, we can ﬁnd a sequence of points (x

) such that

α(x

) and x

go to inﬁnity as n tends to inﬁnity. Then

lim

n→∞

α(x

)

= lim

n→∞



) = 1 − θ

< 1.

Hence, we can ﬁnd some point

x sufﬁciently large, such that α(

x)/

x < 1.

Thus, the Uzawa-Inada condition (UI) is satisﬁed.

Exercise 5.1 In his Economic Dynamics, Baumol (1970) presented a

simple model that captured some of the ideas of “classical” economists

24 Dynamical Systems

formally. Let P

, the net total product in period t, be a function of the

working population L

= F(L

), t ≥ 0,

where F :

→ R

is continuous and increasing, F(0) = 0.

At any time, the working population tends to grow to a size where out-

put per worker is just enough to provide each worker with a “subsistence

level” of minimal consumption M > 0. This is formally captured by the

relation

t+1

Hence,

t+1

F(L

)

≡ α(L

Identify conditions on the average productivity function

F(L)

(L > 0)

that guarantee the following:

(i) There is a unique L

∗

> 0 such that (F(L

∗

)/L

∗

) = M.

(ii) For any 0 < L < L

∗

, the trajectory τ (L) is increasing and con-

verges to L

∗

; for any L > L

∗

, the trajectory τ (L) decreases to L

∗

Example 5.2 Consider an economy (or a ﬁshery) which starts with an

initial stock y (the metaphorical corn of one-sector growth theory or the

stock of renewable resource, e.g., the stock of trouts, the population of

dodos,...). In each period t, the economy is required to consume or

harvest a positive amount c out of the beginning of the period stock y

The remaining stock in that period x

= y

− c is “invested” and the

resulting output (principal plus return) is the beginning of the period

stock y

t+1

. The output y

t+1

is related to the input x

by a “production”

function g. Assume that

[A.1] g :

R → R is continuous, and increasing on R

, g(x) = 0

for x ≤ 0;

[A.2] there is some x

∗

> 0 such that g(x) > xfor0 < x < x

∗

and

g(x) < x for x > x

∗

;

[A.3] g is concave.

1.5 Increasing Laws of Motion 25

The evolution of the system (given the initial y > 0 and the planned

harvesting c > 0) is described by

= y,

= y

− c, t ≥ 0;

t+1

= g(x

) for t ≥ 0.

Let T be the ﬁrst period t, if any, such that x

< 0; if there is no

such t, then T =∞.IfT is ﬁnite we say that the agent (or the resource)

survives up to (but not including) period T . We say that the agent survives

(forever) if T =∞(i.e., if x

≥ 0 for all t).

Deﬁne the net return function h(x) = g(x) − x. It follows that h

satisﬁes

h(x)

≥



0as







0 < x < x

∗

;

x = 0, x

∗

;

x > x

∗

Since g(x) < 0 for all x ≤ 0, all statements about g and h will be

understood to be for nonnegative arguments unless something explicit is

said to the contrary.

Actually, we are only interested in following the system up to the

“failure” or “extinction” time T .

The maximum sustainable harvest or consumption is

H = max

[0,x

∗

]

h(x). (5.3)

We write

t+1

= x

+ h(x

) − c.

If c > H, x

t+1

− x

= h(x

) − c < H − c < 0. Hence, x

will fall below

(extinction) 0 after a ﬁnite number of periods. On the other hand, if

0 < c < H, there will be two roots ξ



and ξ



of the equation

c = h(x),

which have the properties

0 <ξ



<ξ



< x

∗

26 Dynamical Systems

and

h(x) − c

≥



0as









< x <ξ



;

x = ξ



,ξ



;

x <ξ



, x >ξ



We can show that

(a) If x

<ξ



, then x

reaches or falls below 0 in ﬁnite time.

(b) If x

= ξ



, then x

= ξ



for all t.

>ξ



, then x

converges monotonically to ξ



Note that if c = H , there are two possibilities: either ξ



= ξ



(i.e.,

h(x) attains the maximum H at a unique period ξ



) or for all x in a

nondegenerate interval [ξ



,ξ



], h(x) attains its maximum.

The implications of the foregoing discussion for survival and extinc-

tion are summarized as follows.

Proposition 5.4 Let c > 0 be the planned consumption for every period

and H be the maximum sustainable consumption.

(1) If c > H, there is no initial y from which survival (forever) is

possible.

(2) If 0 < c < H , then there is ξ



, with h(ξ



) = c, ξ



> 0

such that survival is possible if and only if the initial stock y ≥ ξ



+ c

(3) c = H implies h(ξ



) = H , and ξ



tends to 0 as c tends to 0.

1.6 Thresholds and Critical Stocks

We now consider some examples of dynamical systems that have been

of particular interest in various contexts in development economics and

in the literature on the management of a renewable resource. It has been

emphasized that the evolution of an economy may depend crucially on

the initial condition (hence, on the “history” that leads to it). It has also

been noted that if the stock of a renewable resource falls below a critical

level, the biological reproduction law may lead to its eventual extinction.

In sharp contrast with the dynamical systems identiﬁed in Proposition

5.1, in which trajectories from positive initial stocks all converge to

a positive ﬁxed point, we sketch some examples where the long-run

behavior of trajectories changes remarkably as the initial condition goes

above a threshold (see Complements and Details).

1.6 Thresholds and Critical Stocks 27

Consider the “no-harvesting” case where S = R

, and the biological

reproduction law is described by a continuous function α :

→ R

Given an initial state x ≥ 0, the state in period t is the stock of the

resources at the beginning of that period and (2.1) is assumed to hold.

When x

= 0, the resource is extinct. We state a general result.

Proposition 6.1 Let S =

and α : S → S be a continuous, increas-

ing function with the following properties [P2]:

[P2.1] α(0) = 0;

[P2.2] there are two positive ﬁxed points x

∗

(1)

∗

(2)

(0 < x

∗

(1)

< x

∗

(2)

)

such that

(i) α(x) < x for x ∈ (0, x

∗

(1)

);

(ii) α(x) > x for x ∈ (x

∗

(1)

, x

∗

(2)

);

(iii) α(x) < x for x > x

∗

(2)

For any x ∈ (0, x

∗

(1)

), the trajectory τ (x) from x is decreasing and

converges to 0; for any x ∈ (x

∗

(1)

, x

∗

(2)

), the trajectory τ (x) from x is

increasing and converges to x

∗

(2)

; for any x > x

∗

(2)

, the trajectory τ (x)

from x is decreasing and converges to x

∗

(2)

Proof. Left as an exercise.

The striking feature of the trajectories from any x > 0 is that these

are all convergent, but the limits depend on the initial condition. Also,

these are all monotone, but again depending on the initial condition –

some are increasing, others are decreasing. We interpret x

∗

(1)

> 0asthe

critical level for survival of the resource.

Exercise 6.1 Consider S =

and α : S → S deﬁned by

α(x) =

+ θ

where θ

,θ

are positive parameters. When θ

> 2

√

, compute the

ﬁxed points of α and verify [P2].

Exercise 6.2 Consider the problem of survival with constant harvesting

discussed in Example 5.2. Work out the conditions for survival with a

28 Dynamical Systems

constant harvest c > 0 when the “production” function g is continuous,

increasing on

, and satisﬁes

[P2.1] g(x) = 0

for x ≤ 0;

[P2.2] there are two positive ﬁxed points x

∗

(1)

, x

∗

(2)

(0 < x

∗

(1)

< x

∗

(2)

)

such that

(i) g(x) < x

for x ∈ (0, x

∗

(1)

);

(ii) g(x) < x

for x ∈ (x

∗

(1)

, x

∗

(2)

);

(iii) g(x) < x

for x > x

∗

(2)

Exercise 6.3 Let S = R

and α : S → S is deﬁned by α(x) = x

, where

m ≥ 1. Show that (i) the ﬁxed points of α are “0” and “1”; (ii) for

all x ∈ [0, 1), the trajectory τ (x) is monotonically decreasing, and con-

verges to 0. For x > 1, the trajectory α(x) is monotonically increasing

and unbounded. Thus, x = 1 is the “threshold” above which sustainable

growth is possible; if the initial x < 1, the trajectory from x leads to

extinction.

Example 6.1 In parts of Section 1.9 we review dynamical systems that

arise out of “classical” optimization models (in which “convexity” as-

sumptions on the preferences and technology hold). Here we sketch an

example of a “nonclassical” optimization model in which a critical level

of initial stock has important policy implications.

Think of a competitive ﬁshery (see Clark 1971, 1976, Chapter 7). Let

(≥0) be the stock or “input” of ﬁsh in period t, and f : R

→ R

the

biological reproduction relationship. The stock x in any period gives rise

to output y = f (x)inthesubsequent period. The following assumptions

on f are introduced:

[A.1] f (0) = 0;

[A.2] f (x) is twice continuously differentiable for x ≥ 0; f



(x) > 0

for x > 0.

[A.3] f satisﬁes the following end-point conditions: f



(∞) < 1 <



(0) < ∞; f



(x) > 0 for x > 0.

[A.4] There is a (ﬁnite) b

> 0, such that (i) f



) = 0; (ii) f



(x) > 0

for 0 ≤ x < b

; (iii) f



(x) < 0 for x > b

In contrast to the present (“nonclassical”) model, the traditional (or

“classical”) framework would replace [A.4] by

1.6 Thresholds and Critical Stocks 29

[A.4



]. f is strictly concave for x ≥ 0(f



(x) < 0 for x > 0), while

preserving [A.1]–[A.3].

In some versions, [A.2] and [A.3] are modiﬁed to allow f



(0) =∞.

In the discussion to follow, we ﬁnd it convenient to refer to a model

with assumptions [A.1]–[A.3] and [A.4



] as classical, and a model with

[A.1]–[A.4] as nonclassical.

We deﬁne a function h (representing the average product function)as

follows:

h(x) = [ f (x)/x] for x > 0; h(0) = lim

x→0

[ f (x)/x]. (6.1)

Under [A.1]–[A.4], it is easily checked that h(0) = f



(0); furthermore,

there exist positive numbers k

∗

k, b

satisfying (i) 0 < b

< b

∗

k < ∞; (ii) f



∗

) = 1; (iii) f (

k) =

k;(iv) f



) = h(b

). Also,

for 0 ≤ x < k

∗

, f



(x) > 1 and for x > k

∗

, f



(x) < 1; for 0 < x <

x < f (x) <

k and for x >

k < f (x) < x; and for 0 < x < b



(x) > h(x) and for x > b

, f



(x) < h(x). Also note that for 0 ≤ x <

, h(x) is increasing, and for x > b

, h(x) is decreasing; for 0 ≤ x < b



(x) is increasing, and for x > b

, f



(x) is decreasing.

A feasible production program from x

> 0 is a sequence (x, y) =

, y

t+1

) satisfying

= x

;0≤ x

≤ y

and y

= f (x

t−1

) for t ≥ 1. (6.2)

The sequence x = (x

)

t≥0

is the input (or stock) program, while the cor-

responding y = (y

t+1

)

t≥0

satisfying (6.2) is the output program. The har-

vest program c = (c

) generated by (x, y) is deﬁned by c

≡ y

− x

for

t ≥ 1. We will refer to (x, y, c) brieﬂy as a program from x

, it being

understood that (x, y) is a feasible production program, and c is the

corresponding harvest program.

A slight abuse of notation: we shall often specify only the stock pro-

gram x = (x

)

t≥0

from x

> 0 to describe a program (x, y, c). It will

be understood that x

= x

;0≤ x

≤ f (x

t−1

) for all t ≥ 1, c

= y

− x

for t ≥ 1.

Let the proﬁt per unit of harvesting, denoted by q > 0, and the rate

of interest γ>0 remain constant over time. Consider a ﬁrm that has an

objective of maximizing the discounted sum of proﬁts from harvesting.

30 Dynamical Systems

A program x

∗

= (x

∗

) of stocks from x

> 0 is optimal if

∞



t=1



(1 + γ )

t−1



∗

≥

∞



t=1



(1 + γ )

(t−1)



for every program x from x

. Write δ = 1/(1 + γ ). Models of this type

have been used to discuss the possible conﬂict between proﬁt maximiza-

tion and conservation of natural resources.

The program (x, y, c) from x

> 0 deﬁned as x

= x

, x

= 0 for t ≥ 1

is the extinction program. Here the entire output f (x

) is harvested in

period 1, i.e., c

= f (x

), c

= 0 for t ≥ 2.

In the qualitative analysis of optimal programs, the roots of the equa-

tion δ f



(x) = 1 play an important role. This equation might not have a

nonnegative real root at all; if it has a pair of unique nonnegative real

roots, denote it by Z ; if it has nonnegative real roots, the smaller one is

denoted by z and the larger one by Z .

The qualitative behavior of optimal programs depends on the value of

δ = 1/(1 +γ ). Three cases need to be distinguished. The ﬁrst two were

analyzed and interpreted by Clark (1971).

Case 1. Strong discounting: δ f



) ≤ 1

This is the case when δ is “sufﬁciently small,” i.e., 1 +γ ≥ f (x)/x

for all x > 0.

Proposition 6.2 The extinction program is optimal from any x

> 0, and

is the unique optimal program if δ f



(

) < 1.

Remark 6.1 First, if δ f



) = 1, there are many optimal programs (see

Majumdar and Mitra 1983, p. 146). Second, if we consider the classical

model (satisfying [A.1]–[A.3] and [A.4



]), it is still true that if δ f



(0) ≤ 1,

the extinction program is the unique optimal program from any x

> 0.

Case 2. Mild discounting: δ f



(0) ≥ 1

This is the case where δ is “sufﬁciently close to 1” (δ>1/ f



(0)) and

Z > b

exists (if z exists, z = 0).

Now, given x

< Z, let M be the smallest positive integer such that

≥ Z; in other words, M is the ﬁrst period in which the pure

1.6 Thresholds and Critical Stocks 31

accumulation program from x

deﬁned by x

= x

, x

t+1

= f (x

) for t ≥ 0

attains Z.

Proposition 6.3 If x

≥ Z, then the program x

∗

= (x

∗

)

t≥0

from x

deﬁned

by x

∗

= x

∗

= Z for t ≥ 1 is the unique optimal program from x

Proposition 6.4 If x

< Z, the program x

∗

= (x

∗

)

t≥0

deﬁned by x

∗

= x

∗

= x

for t = 1,...,M − 1,x

∗

= Z for t ≥ M is the unique optimal

program.

In the corresponding classical model for δ f



(0) > 1, there is a unique

positive K

∗

, solving δ f



(x) = 1. Propositions 6.3 and 6.4 continue to

hold with Z replaced by K

∗

(also in the deﬁnition of M).

Case 3. Two turnpikes and the critical point of departure [δ f



(0) <

1 <δf



)]

In case (i) the extinction program (x, y, c) generated by x

= 0 for all

t ≥ 1 and in case (ii) the stationary program generated by x

= Z for all

t ≥ 0 (which is also the optimal program from Z) serve as the “turnpikes”

approached by the optimal programs. Both the classical and nonclassical

models share the feature that the long-run behavior of optimal programs

is independent of the positive initial stock. The “intermediate” case of

discounting, namely when

1/ f



) <δ<1/ f



(0),

turned out to be difﬁcult and to offer a sharp contrast between the classical

and nonclassical models. In this case

0 < z < b

< b

< Z < k

∗

The qualitative properties of optimal programs are summarized in two

steps.

Proposition 6.5 x

≥ Z, the program x

∗

= (x

∗

)

t≥0

deﬁned by x

∗

= x

∗

= Z for t ≥ 1 is optimal.

A program x = (x

)

t≥0

from x

< Z is a regeneration program if there

is some positive integer N ≥ 1 such that x

> x

t−1

for 1 ≤ t ≤ N , and

= Z for t ≥ N . It should be stressed that a regeneration program

32 Dynamical Systems

may allow for positive consumption in all periods, and need not specify

“pure accumulation” in the initial periods. For an interesting example

of a regeneration program that allows for positive consumption and is

optimal, the reader is referred to Clark (1971, p. 259).

Proposition 6.6 Let x

< Z. There is a critical stock K

> 0 such that

if 0 < x

< K

, the extinction program from x

is an optimal program. If

< x

< Z, then any optimal program is a regeneration program.

In the literature on renewable resources, K

is naturally called the

“minimum safe standard of conservation.” It has been argued that a policy

that prohibits harvesting of a ﬁshery till the stock exceeds K

will ensure

that the ﬁshery will not become extinct, even under pure “economic

exploitation.”

Some conditions on x

can be identiﬁed under which there is a unique

optimal program. But if x

= K

, then both the extinction program and

a regeneration program are optimal. For further details and proofs, see

Majumdar and Mitra (1982, 1983).

1.7 The Quadratic Family

Let S = [0, 1] and A = [0, 4]. The quadratic family of maps is then

deﬁned by

(x) = θ x(1 − x) for (x,θ) ∈ S × A. (7.1)

We interpret x as the variable and θ as the parameter generating the

family.

We ﬁrst describe some basic properties of this family of maps. First,

note that F

(0) = 0(= F

(1)) ∀θ ∈ [0, 4], so that 0 is a ﬁxed point

of F

. By solving the quadratic equation F

(x) = x, one notes that

= 1 − 1/θ is the only other ﬁxed point that occurs if θ>1. For θ>3,

one can show (e.g., by numerical calculations) that the fourth-degree

polynomial equation F

(x): = F

◦ F

(x) has two other solutions (in ad-

dition to 0 and p

). This means that for 3 <θ≤ 4, F

has a period-two

orbit. For θ>1 +

√

6, a new period-four orbit appears. We refer to the

Li–Yorke and Sarkovskii theorems, stated in Section 1.3, for the succes-

sive appearance of periodic points of period 2

(k ≥ 0), as θ increases to a

limit point of θ

≈ 3.57, which is followed by other cascades of periodic