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

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1.9 Some Applications 73

we get in the limit (as t tends to inﬁnity) that

δ f



(

x) = 1.

Hence

x = x

∗

, and this implies

c = c

∗

y = y

∗

. To summarize:

Theorem 9.4 If x

< x

∗

, the optimal program (x

∗

, y

∗

, c

∗

) from x

satisﬁes

∗

> x

∗

t+1

> x

∗

for t ≥ 0, and lim

t→∞

∗

= x

∗

> y

∗

t+1

> y

∗

for t ≥ 1, and lim

t→∞

∗

= y

∗

> c

∗

t+1

> c

∗

for t ≥ 1, and lim

t→∞

∗

= c

∗

Similarly, if x

> x

∗

, the optimal program (x

∗

, y

∗

, c

∗

) from x

satisﬁes

∗

< x

∗

t+1

< x

∗

for t ≥ 0, and lim

t→∞

∗

= x

∗

< y

∗

t+1

< y

∗

for t ≥ 1, and lim

t→∞

∗

= y

∗

< c

∗

t+1

< c

∗

for t ≥ 1, and lim

t→∞

∗

= c

∗

Going back to (9.28), we see that the optimal input program x = (x

∗

)

t≥0

is described by

∗

t+1

= i( f (x

∗

))

≡ α(x

∗

) (9.30



)

where α :

→ R

is the composition map i · f.We shall often refer

to α as the optimal transition function.

Exercise 9.3 Using [F.4] and Lemma 9.4 show that α is increasing.

Show that for x < x

∗

, α(x) > x and for x > x

∗

, α(x) < x (recall

Example 5.1).

Exercise 9.4 Consider the dynamic optimization problem:

maximize

∞



t=0

log c

subject to c

t+1

+ x

t+1

= Ax

≥ 0, x

≥ 0, for t ≥ 0,

74 Dynamical Systems

where A > 0, 0 <β<1, 0 <δ<1, and x

= x

is given.

(a) Show that there is a unique optimal program which is interior.

(b) Show that the optimal investment policy function i(y) and the

optimal consumption policy function c(y) are both linear:

i(y) = δβy,

c(y) = (1 − δβ)y.

Hence, the optimal input program is described by

t+1

= (Aδβ)x

, x

> 0.

Exercise 9.5 (Optimal growth in a productive economy) Consider the

one-sector aggregative model of optimal growth with discounting:

maximize

∞



t=0

u(c

)

subject to c

+ x

≤ y

+ x

≤ y

= f (x

t−1

) for t ≥ 1;

> 0 given.

Assume that

(i) u(c) = c

for all c ≥ 0 and α ∈ (0, 1);

(ii) f :

−→ R

is continuous, concave, and strictly increasing;

(iii) f is differentiable and f



(x) > 0 on R

;

(iv) f (0) = 0, f



(0) > 1.

Given the utility function, one easily checks that u is continuous,

strictly concave and strictly increasing, and differentiable on

. Fur-

thermore, lim

c→0



given.

Let the asymptotic productivity be deﬁned by

γ = lim

x→∞



f (x)



= lim

x→∞



(x).

Lemma 9.5.1 If δ<γ

−a

, then for any given y

> 0 there exists B ∈

such that every feasible consumption program (c

)

∞

t=0

from initial

stock y

satisﬁes



∞

t=0

u(c

) < B.

1.9 Some Applications 75

Proof. If γ<1, then the proof is direct. So, conﬁne attention to a

situation where γ ≥ 1 and a > 0. Fix y

> 0. Deﬁne the pure accu-

mulation program

y = (y

) where y

= f (y

t−1

), ∀t ≥ 1, y

= y

. It is

easy to check that every feasible consumption program from initial stock

satisﬁes c

≤ y

. Let B =



∞

t=0

u(y

). It is sufﬁcient to show that

B < ∞. Note that

y = (y

) is monotone increasing, so it must diverge

to plus inﬁnity if it is not bounded. If (

) is bounded, clearly B < ∞.

Observe that

t+1

f (

)

−→ γ as t −→ ∞

Since δ<γ

−a

, there exists β>1, > 0 such that

δ<(βγ)

−(a+)

There exists T such that for t ≥ T,

> 1

t+1

<βγy

so that, in particular,

T +k

< (βγ)

, k ≥ 1,

and

T +k

) ≤ (y

T +k

)

a+

≤ [(βγ)

]

a+

, k ≥ 0.

Thus,

∞



t=0

u(y

) =

T −1



t=0

u(y

) + δ



∞



k=0

u(y

T +k

)



≤

T −1



t=0

u(y

) + δ

)

a+

∞



k=0

[δ(βγ)

a+

]

< ∞, since δ<(βγ)

−(a+)

The proof is complete.

Remark 9.5.1 There exists a unique optimal program. Every optimal

program (x

∗

, c

∗

, y

∗

) is strictly positive. The value function V (y) is strictly

76 Dynamical Systems

increasing, strictly concave, differentiable on R

and V



) = u



Further, the following Ramsey-Euler equation holds:



∗

) = δu





∗

t+1







∗





∗

) = δV





∗

t+1







∗



Remark 9.5.2 The optimal consumption and optimal investment func-

tions are increasing.

Remark 9.5.3 Every optimal program (x

∗

, c

∗

, y

∗

) is monotonic.

Remark 9.5.4 If f



(0) <

, every optimal program converges to zero.

For the simplicity of notation, we drop the superscript ∗ in the next

two results when we deal with optimal programs.

Remark 9.5.5 If f



(0) >

, then every optimal program is bounded

away from zero.

Hint . It is sufﬁcient to show that there exists ε>0 such that for each

initial stock y

∈ (0,ε), the optimal y

> y

To see this, suppose not.

Then, there exists a sequence of strictly positive initial stocks {y

}↓

0 such that the next period’s optimal stock y

= f (x

) ≤ y

. Using

concavity of the value function, we have V



) ≥ V



). However,

for n large enough, δ f



) > 1 so that the Ramsey-Euler equation

implies



) = δV













> V







a contradiction.

Remark 9.5.6 If δ>γ

−1

, then every optimal program of consumption,

input, and output diverges to inﬁnity.

Hint. Fix any y

> 0. As the optimal input program {x

} is monotonic,

if it does not diverge to inﬁnity, it must converge to either zero or

a strictly positive stationary optimal stock. However, δ>γ

−1

implies

δ f



(x) > 1, ∀x ≥ 0, which in turn implies (using Proposition 9.8) that

} is bounded away from zero and, furthermore, there does not exist a

1.9 Some Applications 77

strictly positive stationary optimal stock. Thus, {x

}→∞ and, there-

fore, {y

}→∞. Finally, to see that the optimal consumption {c

}→∞,

suppose not. Then, {c

} is bounded above. Using the strict concavity of

u and the Ramsey-Euler equation,



) = δu



t+1

) f



Since δ f



(x) > 1, ∀x ≥ 0, we have that {c

} is an increasing sequence

which must then converge to some c



> 0. Taking limits on both sides of

the Ramsey-Euler equation, we obtain u



) = δγ u



), i.e., δ = γ

−1

contradiction.

1.9.5 Optimization with Wealth Effects: Periodicity and Chaos

In the optimal growth model of Section 1.9.4.1, the utility (or return)

function u is deﬁned on consumption only. The need for a more general

framework in which the utility function depends on both consumption

and input has been stressed in the theory of optimal growth as well as the

theory of natural resource management. A step in this direction – even in

the aggregative or one good model – opens up the possibility of periodic

or chaotic behavior of the optimal program. In this section, we sketch

some examples from Majumdar and Mitra (1994a,b) (see Complements

and Details).

1.9.5.1 Periodic Optimal Programs

We consider an aggregative model, speciﬁed by a production function

f :

→ R

,awelfare function w : R

→ R

, and a discount factor

δ ∈ (0, 1).

The following assumptions on f are used:

[F.1] f (0) = 0; f is continuous on

[F.2] f is nondecreasing and concave on

[F.3] There is some K > 0 such that f (x) > xfor0 < x < K, and

f (x) < x when x > K.

We deﬁne a set ⊂

as follows:

={(x, z) ∈

: z ≤ f (x)}. (9.31)

78 Dynamical Systems

The following assumptions on w are used:

[W.1] w(x, c) is continuous on

[W.2] w(x, c) is nondecreasing in x given c, and nondecreasing in c,

given x on

. Further, if x > 0, w(x, c) is strictly increasing in

con.

[W.3] w(x, c) is concave on

. Furthermore, if x > 0, w(x, c) is

strictly concave in c on .

A program x = (x

)

∞

from x

≥ 0 is a sequence satisfying

= x

, 0 ≤ x

t+1

≤ f (x

) for t ≥ 0.

The consumption program c = (c

t+1

)

∞

is given by

t+1

= f (x

) − x

t+1

for t ≥ 0.

As in Section 1.9.5.1, (x

, c

t+1

)

t≥0

≤ B ≡ max(K , x

) for any program

x from x

≥ 0, we have (x

, c

t+1

)

t≥0

≤ B ≡ max(K , x

) for t ≥ 0. In par-

ticular, if x

∈ [0, B], then x

, c

t+1

≤ B for t ≥ 0.

A program x

∗

=(x

∗

)

∞

from x

≥ 0isoptimal if

∞



t=0

w(x

∗

, c

∗

t+1

) ≥

∞



t=0

w(x

, c

t+1

)

for every program x from x

A program x from x

is stationary if x

= x for t ≥ 0. It is a stationary

optimal program if it is also an optimal program from x

. In this case, x

is called a stationary optimal stock. (Note that 0 is a stationary optimal

stock.) A stationary optimal stock x

is nontrivial if x

> 0. We turn ﬁrst

to the possibility of a periodic optimal program.

Example 9.1 Deﬁne f :

→ R

f (x) =



(32/3)x − 32x

+ (256/3)x

for 0 ≤ x < 0.25

1 for x ≥ 0.25

Deﬁne w :

→

w(x, c) = 2x

1/2

for all (x, c) ∈ R

Finally, deﬁne

δ = (1/3).

1.9 Some Applications 79

It can be checked that f satisﬁes [F.1]–[F.3] with B = 1 and w satisﬁes

[W.1]–[W.3].

Now, let δ be any discount factor satisfying

δ<δ<1. We ﬁx this δ

in what follows. Deﬁne an open interval

A(δ) ={x:0.25 < x < 3δ

/(1 + 3δ

)}

and also a point

x(δ) = δ/(1 +δ)

It can be shown then that

(a) x(δ) is the unique non-trivial stationary optimal stock (note that

x(δ) ∈ A(δ));

(b) for every x

∈ A(δ), with x

= x(δ), the optimal program x

∗

)

∞

from x

is periodic with period 2.

1.9.5.2 Chaotic Optimal Programs

We consider a family of economies indexed by a parameter θ (where

θ ∈ I ≡ [1, 4]). Each economy in this family has the same production

function (satisfying [F.1]–[F.3]) and the same discount factor δ ∈ (0, 1).

The economies in this family differ in the speciﬁcation of their utility or

one-period return function w :

× I → R

(w depending explicitly

on the parameter θ). For a ﬁxed θ ∈ [1, 4], the utility function w(·,θ)

can be shown to satisfy [W.1]–[W.3].

Example 9.2 The numerical speciﬁcations are as follows:

f (x) =



(16/3)x − 8x

+ (16/3)x

for x ∈ [0, 0.5)

1 for x ≥ 0.5

δ = 0.0025. (9.32)

The function w is speciﬁed in a more involved fashion. To case the

writing denote L ≡ 98, a ≡ 425. Also, denote by S the closed interval

[0, 1] and by I the closed interval [1, 4], and deﬁne the function m :

S × I → S by

m(x,θ) = θ x(1 − x) for x ∈ S, θ ∈ I

80 Dynamical Systems

and u : S

× I → R by

u(x, z,θ) = ax − 0.5Lx

+ zm(x,θ) − 0.5z

−δ{az − 0.5Lz

+ 0.5[m(z,θ)]

}. (9.33)

Deﬁne a set D ⊂ S

D ={(x, c) ∈

× S: c ≤ f (x)}

and a function w : D × I →

w(x, c,θ) = u(x, f (x) − c,θ) for (x, c) ∈ D and θ ∈ I. (9.34)

We now extend the deﬁnition of w(·,θ) to the domain .For

(x, c) ∈with x > 1 (so that f (x) = 1 and c ≤ 1), deﬁne

w(x, c,θ) = w(1, c,θ). (9.35)

Finally, we extend the deﬁnition of w(·,θ) to the domain

.For

(x, c) ∈

with c > f (x), deﬁne

w(x, c,θ) = w(x, f (x),θ). (9.36)

One of the central results in Majumdar and Mitra (1994b) is the

following:

Proposition 9.4 The optimal transition functions for the family of

economies ( f,w(·,θ),δ) are given by

(x) = θ x(1 − x) for all x ∈ S. (9.37)

In other words, for a ﬁxed

θ, the optimal input program is described

by x

∗

t+1

θ x

∗

(1 − x

∗

1.9.5.3

Robustness of Topological Chaos

In Example 9.2, we provide a speciﬁcation of ( f,w,

δ) for which topo-

logical chaos is seen to occur. A natural question to ask is whether this

is fortuitous (so that if the parameters f , w,orδ were perturbed ever

so little, the property of topological chaos would disappear) or whether

this is a “robust” phenomenon (so that small perturbations of f , w,orδ

would preserve the property of topological chaos).

We proceed now to formalize the above question as follows. Deﬁne

F ={f : R

→ R

satisfying [F.1]–[F.3]}; W ={w : R

→ R

sat-

isfying [W.1]–[W.3]};  ={δ:0<δ<1}.Aneconomy e is deﬁned by

1.9 Some Applications 81

a triple ( f,w,δ) ∈ F × W × . The set of economies, F × W ×  is

deﬁned by E.

Consider the economy e = ( f,w,

δ) deﬁned in Example 9.3 where

θ = 4. We would like to demonstrate that all economies e ∈ E “near”

the economy e will exhibit topological chaos. Thus, the property of topo-

logical chaos will be seen to persist for small perturbations of the original

economy.

A convenient way to make the above idea precise is to deﬁne for e ∈ E,

the “distance” between economies e and e,by

d(e, e) = sup

x≥0

f (x) − f (x)

+ sup

(x ,c)≥0

w(x, c) − w(x, c)

+|δ −

δ|.

Note that d(e, e) may be inﬁnite.

Before we proceed further, we have to clarify two preliminary points.

First, if we perturb the original economy e (i.e., choose another econ-

omy e = e), we will, in general, change the set of programs from any

given initial stock, as also the optimal program from any given initial

stock. That is, programs and optimal programs (and hence optimal pol-

icy functions) are economy speciﬁc. Thus, given an economy e,weuse

the expressions like “e-program,” “e-optimal program,” and “e-optimal

transition function” with the obvious meanings.

Second, recall that for the original economy e, K = 1 (recall [F.3]), and

so if the initial stock was in [0, 1], then for any program from the initial

stock, the input stock in every period is conﬁned to [0.1]. Furthermore, for

any initial stock not in [0, 1], the input stock on any program belongs to

[0, 1] from the very next period. In this sense ([0, 1,α

)) is the “natural”

dynamical system for the economy e. When we perturb the economy, we

do not wish to restrict the kind of perturbation in any way, and so we would

have to allow the new economy’s production function to satisfy [F.3] with

a K > 1. This changes the “natural” state space of the dynamical system

for the new economy. However, recalling that we are only interested

in “small” perturbations, it is surely possible to ensue that d(e, e) ≤ 1,

so that we can legitimately take the “natural” state space choice to be

J = [0, 2].

We can now describe our result (from Majumdar and Mitra 1994a)

formally as follows.

Proposition 9.5 Let e = ( f,w,

δ) be the economy described in Example

9.2. There exists some ε>0 such that for every economy e ∈ E with

82 Dynamical Systems

d(e, e) <ε, the dynamical system (J,α) exhibits Li–Yorke chaos, where

a is the e-optimal transition function and J = [0, 2].

Example 9.3 (Optimal exploitation of a ﬁshery) We return to the model

of a ﬁshery introduced in Example 6.1 and, following Dasgupta (1982),

sketch a dynamic optimization problem in which the return function

depends on both the “harvest” and the “stock.”

The resource stock (biomass of the ﬁsh species) at the end of period t is

denoted by x

. It generates y

t+1

, the stock in the next period, according to

f , the biological reproduction function (also called the stock recruitment

function):

t+1

= f (x

If c

t+1

is harvested in period (t + 1), then the stock at the end of time

period (t + 1) is x

t+1

, given by

t+1

= f (x

) − c

t+1

It is usual to assume that f (0) = 0, f is continuous, nondecreas-

ing, and concave, with lim

x→0

[ f (x)/x] > 1 and lim

x→∞

[ f (x)/x] = 0

(although there are important variations, especially dealing with noncon-

cave functions, f ).

We can ﬁnd B > 0 such that whenever x

is in [0, B], x

t+1

is also in

[0, B]. Thus [0, B] is a legitimate choice for the state space, S.

Harvesting the ﬁshery is not costless. Speciﬁcally, the harvest, c

t+1

depends on the labor effort, e

t+1

. We express the effort e

t+1

required to

harvest c

t+1

when the biomass available for harvest is y

t+1

= H(c

t+1

, y

t+1

with H increasing in its ﬁrst argument and decreasing in its second. This

is the cost of exploiting the ﬁshery. The beneﬁt obtained depends on the

harvest level c

t+1

B(c

t+1

) = pc

t+1

where p > 0 is the price (assumed constant over time) per unit of the

ﬁsh. The return in period (t + 1) is then the beneﬁt minus the cost:

w(c

t+1

, x

) = B(c

t+1

) − H (c

t+1

, f (x

))