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

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1.9 Some Applications 53

1.9.3 Balanced Growth and Multiplicative Processes

Notation For two square (n × n) matrices A = (a

) and B = (b

write

A ≥ B if a

≥ b

for all i, j;

A > B if A ≥ B and A = B;

A >> B if a

> b

for all i, j.

A is nonnegative (resp. positive, strictly positive)ifA ≥(resp. >, >>)

Primed letters denote transposes. When A is a square matrix,

= TAT

−1

denotes the transform of A by the nonsingular n × n

matrix T .

Consider the dynamical system (S,α) where S =

and α : S → S

is deﬁned by

α(x) = Ax , (9.15)

where A ≥ 0isanonnegative n × n matrix. The dynamical system (9.15)

leads to the study of

= A

, (9.16)

where x

≥ 0. We shall ﬁrst prove a fundamental result on nonnegative

indecomposable matrices. A permutation matrix is obtained by permut-

ing the columns of an identity matrix.

An n × n matrix A(n ≥ 2) is said to be indecomposable if for no

permutation matrix  does A

= A

−1

= (

) where A

, A

are square. (A

−1

is obtained by performing the same permutation on

the rows and on the columns of A.)

Proposition 9.2 Let A ≥ 0 be indecomposable. Then

(1) A has a characteristic root λ>0 such that

(2) to λ can be associated an eigenvector x

>> 0;

(3) if β is any characteristic root of A, |β|≤λ;

(4) λ increases when any element of A increases;

(5) λ is a simple root of φ(t) ≡ det(tI − A) = 0

54 Dynamical Systems

Proof.

(1) (a) If x > 0, then Ax > 0. For if Ax = 0, A would have a column

of zeros, and so would not be indecomposable.

(1) (b) A has a characteristic root λ>0.

Let  ={x ∈

| x > 0,



= 1}.Ifx ∈ , we deﬁne T (x) =

[1/ρ(x)]Ax where ρ(x) > 0 is so determined that T (x) ∈  (by (a) such

a ρ exists for every x ∈ ). Clearly, T (x) is a continuous transformation

of S into itself, so, by Proposition 2.2, there is an x

∈  with x

T (x

) = [1/ρ(x

)]Ax

. Put λ = ρ(x

(2) x

>> 0. Suppose that after applying a proper ,





,ξ>>0.

Partition A

accordingly. A

= λ

yields









λξ



;

thus A

ξ = 0, so A

= 0, violating the indecomposability of A.

If M = (m

) is a matrix, we henceforth denote by M

∗

the matrix

∗

= (|m

|).

(3–4) If 0 ≤ B ≤ A, and if β is a characteristic root of B, then |β|≤

λ. Moreover, |β|=λ implies B = A.



is indecomposable and therefore has a characteristic root λ

> 0

with an eigenvector x

>> 0: A



= λ

. Moreover, β y = By. Taking

absolute values and using the triangle inequality, we obtain

(i) |β| y

∗

≤ By

∗

≤ Ay

∗

.So

(ii) |β| x



∗

≤ x



∗

= λ



∗

Since x

>> 0, x



∗

> 0, thus |β|≤λ

Putting B = A one obtains |β|≤λ

. In particular λ ≤ λ

and since,

similarly, λ

≤ λ, λ

= λ.

Going back to the comparison of B and A and assuming that |β|=λ,

one gets from (i) and (ii)

λy

∗

= By

∗

= Ay

∗

From λy

∗

= Ay

∗

, application of (2) gives y

∗

>> 0. Thus By

∗

= Ay

∗

together with B ≤ A yields B = A.

1.9 Some Applications 55

(5) (a) If β is a principal submatrix of A, and β is any characteristic

root of B, |β| <λ.

β is also a characteristic root of the matrix

B = (

). Since A is

indecomposable,

B < A

for a proper . Hence |β| <λby (3–4).

(5) (b) λ is a simple root of φ(t) = det(tI − A) = 0. φ



(λ)isthe

sum of the principal (n − 1) ×(n − 1) minors of det(λI − A). Let A

be one of the principal (n − 1) × (n − 1) submatrices of A. By (5) (a),

det(tI − A) cannot vanish for t

 λ. Hence, det(λI − A

) > 0 and



(λ) > 0.

As an immediate consequence of (4), one obtains

min



≤ λ ≤ max



and one equality holds only if all row sums are equal (then they both

hold).

This is proved by increasing (respectively decreasing) some elements

of A so as to make all row sums equal to

max





resp. min





A similar result naturally holds for column sums.

We write λ(A) to denote the (Perron–Frobenius or dominant) charac-

teristic root whose existence is asserted in Proposition 9.2.

Remark 9.1

(a) λ(A) = λ(A



), λ(θ A) = θλ( A) for θ>0; λ(A

) = (λ( A))

for any

positive integer k.

(b) Any eigenvector y ≥ 0 associated with λ(A) is, in fact, strictly

positive, i.e., y >> 0 and y = θ x

for some θ>0.

The deﬁnitive result on the long-run behavior of (9.16) is the

following.

Proposition 9.3 Let A be an indecomposable nonnegative matrix, and

λ = λ( A).

(i) lim

t→+∞

(A/λ)

exists if and only if there is a positive integer k

such that A

>> 0.

56 Dynamical Systems

(ii) In the case of the convergence of (A/λ)

, every column of the limit

matrix is a strictly positive column eigenvector of A and every row is a

strictly positive row eigenvector of A , both associated with λ.

Proof. Proposition 9.2 ensures that λ>0, so that ( A/λ) makes sense.

Moreover, the dominant root of ( A/λ) is unity. This enables us to assume,

without loss of generality, that λ(A) = 1.

First we shall prove (ii). To this end, let B = lim A

. Clearly, B ≥ 0.

We also have lim A

t+1

= B, so that AB = A lim A

= lim A

t+1

= B

and, similarly, BA = B. AB = B implies that Ab = b for any column

of B. It follows, by Remark 9.1, that b is either a strictly positive col-

umn eigenvector of A associated with λ = 1 or the column zero vector.

Likewise BA = B implies that any row of B is either a strictly positive

row eigenvector of A associated with λ = 1 or the row zero vector. In

fact, B = (b

) is a strictly positive matrix. For if some b

= 0, the above

results imply that both the i th row and the jth column of B vanish, and

hence, all the rows and columns vanish. But the possibility of B = 0is

ruled out, because Bx

= lim A

= x

for a strictly positive column

eigenvector x

of A associated with λ = 1. This proves (ii).

The “only if ” part of (i) immediately follows from (ii). In fact, if

lim A

= B exists, B must be strictly positive by (ii). Hence A

must also

be strictly positive for t ≥ k, whenever k is large enough. For proof of the

“if ” part of (i), ﬁrst assume that the assertion is true for any strictly pos-

itive matrix. Then, since A

>> 0 and λ(A

) = λ( A)

= 1 by Remark

9.1, lim(A

)

= B >> 0 exists. Every positive integer t is uniquely ex-

pressible in the form t = p(t)k + r(t), where p(t), r(t) are nonnegative

integers with 0 ≤ r(t) < k, lim p(t) =+∞. Therefore, letting (A

)

p(t )

B + R(t), we have lim R(t) = 0. Hence A

= A

r(t )

)

p(t )

= A

r(t )

(B +

R(t)) = A

r(t )

B + A

r(t )

R(t) → B because A

r(t )

B = B and A

r(t )

bounded. The reason for A

r(t )

B = B is that since A and A

have eigen-

vectors in common, the columns of B are strictly positive column eigen-

vectors of A associated with λ = 1; hence A

r(t )

B = AA···AB = B.

Thus it remains to prove the assertion for strictly positive matrices.

Now let A be a strictly positive matrix with λ(A) = 1. To prove the

convergence of A

, we only have to show that the vector sequence {A

is convergent for any positive y > 0.

Set y

= A

y for any given y > 0. y

is a solution of the difference

equation

t+1

= Ay

, y

= y.

1.9 Some Applications 57

Take a strictly positive column eigenvector x = (x

)ofA associated with

λ = 1, and let

i,t

= y

i,t

= min[θ

i,t

,...,θ

n,t

= max[θ

i,t

,...,θ

n,t

We let l(t) be an integer for which m

= θ

l(t),t

. Then, for any i

i,t+1



j=1

j,t



j=1

(θ

j,t

= m

il(t)

l(t)



j=l(t)

j,t

≤ m

il(t)

l(t)

+ M



j=l(t)

= (m

− M

il(t)

l(t)

+ M



j=1

= (m

− M

il(t)

l(t)

+ M

Now let

δ = min

i, j

,ε= min

i, j

Then δ>0, ε>0, and the above relations, if divided by x

, become

i,t+1

≤ (m

− M

)εδ + M

(i = 1,...,n),

from which we get

t+1

≤ (m

− M

)εδ + M

which can be rearranged to

εδ(M

− m

) ≤ M

− M

t+1

Similarly, we have

εδ(M

− m

) ≤ m

t+1

− m

58 Dynamical Systems

Since M

≥ m

, the above two inequalities imply that {m

}is nondecreas-

ing and bounded above, and {M

} is nonincreasing and bounded below.

Hence lim m

= m and lim M

= M exist. Then, either of the above two

inequalities entails m = M. In view of the deﬁnition of m

, M

,wehave

thereby proved lim y

= mx.

Remark 9.2 An economic implication of the above result is the relative

stability of a balanced-growth solution x

= λ

x, λ = λ(A), λx = Ax of

the self-sustained system:

t+1

= Ax

(9.17)

where A is a nonnegative matrix such that A

>> 0 for some k. For any

solution x

= A

of the equation, x

/λ

converges to Bx

,ast tends

to inﬁnity, where B = lim(A/λ)

. Stated differently, all the component

ratios x

i,t

/λ

converge to a common limit. The latter statement can

be justiﬁed in the following way: x

/λ

is a solution of y

t+1

= (A/λ)y

with λ(A/λ) = 1, so that x

i,t

/λ

corresponds to θ

i,t

in the proof of

Proposition 9.3, which converges to a limit common to all i.

Remark 9.3 A nonlinear multiplicative process, ﬁrst studied by Solow

and Samuelson (1953) was explored in a number of contexts in economics

and mathematical biology. We sketch the simplest case.

Take S =



and let α : S → S satisfy:

[A.1] α is continuous;

[A.2] α is homogeneous of degree 1, i.e., α(µx) = µα(x) for µ>0

and x ≥ 0;

[A.3] “x



> x” implies α(x



) >> a(x).

One can consider a large population consisting of  types of individuals

(e.g., the population can be divided into  age groups) or a community

consisting of  interacting species (see Demetrius (1971) for a review of

examples of this type). One can also think of x

i,t

as the stock of good

i in period t. Then the interpretation of the mathematical model as a

description of a dynamic multisector economy was that

we suppose all optimization and allocation problems to have been solved in

one way or the other. We do not question what happens inside the sausage

grinder: we simply observe that inputs ﬂow into the economy and outputs

appear. Since the economy is assumed to be closed . . . , all outputs become

1.9 Some Applications 59

the inputs in the next period. Thus, it is as if we had a single production

process

permitting continuous substitution of inputs (Solow and Samuelson 1953,

p. 412).

One can then show that there are λ>0 and a unique (in ) v = (v

such that

λv

= α

(v) for i = 1,...,.

Moreover, let x

be the trajectory from x

>> 0. Then lim

t→∞

i,t

/(v

) = c > 0, where c is independent of i. Signiﬁcant progress

has been made to replace [A.3] with weaker assumptions (see Nikaido

1968).

1.9.4 Models of Intertemporal Optimization with

a Single Decision Maker

We now turn to some inﬁnite horizon economic models of dynamic op-

timization with a single decision maker. These give rise to dynamical

systems that describe the evolution of optimal states or actions. In what

follows, we use some of the concepts and insights from the literature on

dynamic programming. In Chapter 6 we explore a more general prob-

lem of discounted dynamic programming under uncertainty, and prove

a few deep results by appealing to the contraction mapping theorem. In

contrast, in this section we rely on rather elementary methods to char-

acterize optimal actions and to touch upon some fundamental issues on

the role of prices in decentralization. We shall now informally sketch a

model of deterministic dynamic programming (with discounting). It is

a useful exercise to spell out formally how this framework encompasses

the speciﬁc cases that we study in detail.

A deterministic stationary dynamic programming problem (SDP) is

speciﬁed by a triple < S, A,ϕ, f, u,δ >where:

– S is the state space with generic element s; we assume that S is a

nonempty complete separable metric space (typically, a closed subset



– A is the set of all conceivable actions, with generic element. We assume

that A is a compact metric space.

– ϕ : S → A is a set valued mapping: for each s ∈ S, the nonempty

set ϕ(s) ⊂ A describes the set of all feasible actions to the decision

60 Dynamical Systems

maker, given that the state he observes is (s) (some continuity prop-

erties of ϕ are usually needed, see Chapter 6, especially Section 6.3).

– f : S × A → S is the law of transition for the system: for each current

state–action pair (s, a) the state in the next period is given by f (s) ∈ S

(again, f typically satisﬁes some continuity property).

– u : S × A → R is the immediate return (or one-period utility) function

that speciﬁes a return u(s, a) immediately accuring to the decision

maker when the current state is s, and the action a is chosen (some

continuity and boundedness properties are usually imposed on u).

– δ ∈ [0, 1) is the discount factor.

The sequential decision-making process is described as follows: An

initial state s

∈ S is given, and if the decision maker selects an action

in the set ϕ(s

) of actions feasible to him or her, two things happen.

First, an immediate return u(s

, a

) is generated, and secondly, the system

moves to a state s

= f (s

, a

), which will be observed tomorrow. The

story is repeated. Now, when the action a

is chosen in ϕ(s

) in period

t, the present value of the immediate return u(s

, a

) in period t is given

by δ

u(s

, a

). Given s

∈ S, the aim of the decision maker is to choose

a sequence (a

) of action that solves

maximize

∞



t=0

u(s

, a

)

subject to s

t+1

= f (s

, a

), t = 0, 1 ...

∈ φ(s

), t = 0, 1, 2,...

1.9.4.1 Optimal Growth: The Aggregative Model

In this subsection we review several themes that have been central to

the literature on optimal economic development. We use an aggrega-

tive, discrete time, “discounted” model (in contrast with the pioneering

“undiscounted” continuous time model of Ramsey (1928)), and deal with

the following questions:

– the existence of an optimal program;

– the “duality” theory of dynamic competitive prices (attainment of

the optimal program in an informationally decentralized economy by

using competitive prices);

– optimal stationary programs (“golden rules”);

1.9 Some Applications 61

– asymptotic or long-run properties of optimal programs (“turnpike

theorems”).

Consider an inﬁnite horizon, one-good (“aggregative”) model with

a stationary technology described by a gross output function f , which

satisﬁes the following assumptions [F]:

[F.1] f :

→ R

is continuous on R

[F.2] f (0) = 0.

[F.3] f (x) is twice differentiable at x > 0, with f



(x) > 0 and



(x) < 0.

[F.4]

(a) lim

x↓0



(x) =∞; lim

x↑∞



(x) = 0;

(b) f (x) →∞ as x →∞.

Note that the assumptions imply that f is strictly increasing and strictly

concave on

Corresponding to any f satisfying [F.1]–[F.4], there exists a unique

number K > 0, such that

(i) f (K) = K,

(ii) f (x) > x for 0 < x < K,

(iii) f (x) < x for x > K.

We refer to K as the maximum sustainable stock for the technology

described by f .

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. (9.18)

The sequence x = (x

)

t≥0

is the input program, while the corresponding

y = (y

t+1

)

t≥0

satisfying (9.18) is the output program. The consumption

program c = (c

), generated by (x, y), is deﬁned by

≡ y

− x

for t ≥ 1. (9.19)

We 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

consumption program.

A program (x, y, c) from x

is called strictly positive (written

(x, y, c) >> 0) if x

> 0, y

t+1

> 0, c

t+1

> 0 for t ≥ 0. It is a standard

62 Dynamical Systems

exercise to check that for any program (x, y, c) from x

we have

, y

t+1

, c

t+1

) ≤ (B, B, B) for t ≥ 0, where B = max(x

, K).

Again, a slight abuse of notation: we shall often specify only the input

program x = (x

)

t≥0

from x

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

be understood that x

= x

and 0 ≤ x

≤ f (x

t−1

) for all t ≥ 1, and (9.18)

and (9.19) hold. Indeed we shall also refer to (x, y, c) as a program

from y

> 0 to mean that it is really a program from the unique x

> 0

such that y

= f (x

). Note that decisions on “consumption today versus

consumption tomorrow” begin in period 1.

A social planner evaluates alternative programs according to some

welfare criterion. The criterion we examine is a “discounted sum of

utilities generated by consumptions.” More precisely, let u :

→ R

be the utility (or return) function and δ be the discount factor satis-

fying 0 <δ<1. In what follows, u satisﬁes the following assump-

tions [U]:

[U.1] u is continuous on

[U.2] u(c) is twice continuously differentiable at c > 0 with u





c↓0



Clearly, [U.1] and [U.2] imply that u is strictly concave and increasing

. We normalize u(0) = 0.

For any program (x, y, c) from x

> 0, one has

0 ≤ u(c

) ≤ u(B) for all t ≥ 1.

A program (x

∗

, y

∗

, c

∗

) from x

> 0isoptimal if

∞



t=1

t−1



u(c

) − u



∗



≤ 0

for all programs (x, y, c) from x

> 0.

It is useful to recast the optimization problem just outlined in a dynamic

programming framework. One can take S = [0, B], and A = [0, 1] ≡

ϕ(s) for all s ∈ S.

We ﬁrst deal with the question of existence of an optimal program.

Theorem 9.1 There is a unique optimal program from any x

> 0.