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

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1.9 Some Applications 63

Proof. Let b ≡ u(B). If (x, y, c) is any program from x

, then 0 ≤

u(c

) ≤ b, for t ≥ 1. So for every T ≥ 1



t=1

t−1

u(c

) ≤ b/(1 − δ).

Hence,

∞



t=1

t−1

u(c

) ≡ lim

T →∞



t=1

t−1

u(c

) ≤ b/(1 − δ).

Let

a = sup



∞



t=1

t−1

u(c

):(x, y, c) is a program from x



Choose a sequence of programs (x

, y

, c

) from x

such that

∞



t=1

t−1





≥ a − (1/n) for n = 1, 2, 3,... (9.20)

Using the continuity of f , and the diagonalization argument, there is a

program (x

∗

, y

∗

, c

∗

) from x

, and a subsequence of n (retain notation)

such that for t ≥ 0,



, y

t+1

, c

t+1



−→



∗

, y

∗

t+1

, c

∗

t+1



as n →∞. (9.21)

We claim that (x

∗

, y

∗

, c

∗

) is an optimal program from x

. If the claim is

false, then we can ﬁnd ε>0 such that

∞



t=1

t−1



∗



< a − ε.

Pick T such that δ

[(b/1 − δ)] < (ε/3). Using (9.20), and the continuity

of u, we can ﬁnd

n, such that for all n ≥



t=1

t−1







t=1

t−1



∗



+ (ε/3).

64 Dynamical Systems

Thus, for all n ≥

n we get

∞



t=1

t−1





≤



t=1

t−1





+ δ

[(b/1 − δ)]



t=1

t−1





+ (ε/3)



t=1

t−1



∗



+ (2ε/3) ≤

∞



t=0

t−1



∗



+ (2ε/3)

< a − (ε/3).

But this leads to a contradiction to (9.19) for n > max[

n, (3/ε)]. This

establishes our claim. Uniqueness of the optimal program follows from

the strict concavity of u and f .

Note that for any x

> 0, the unique optimal program (x

∗

, y

∗

, c

∗

) from

is strictly positive. For any y > 0, deﬁne the value function V : R

→

V (y

) =

∞



t=1

t−1



∗



where (x

∗

, y

∗

, c

∗

) is the unique optimal program from x

= f

−1

). We

see V (0) = 0. For any y

> 0, deﬁne the optimal consumption policy

function c :

→ R

c(y

) = c

∗

where (x

∗

, y

∗

, c

∗

) is the unique optimal program from y

(or, more for-

mally, from x

= f

−1

)).

We now state some of the basic properties of the value and consumption

policy functions.

Remark 9.4 Let (x

∗

, y

∗

, c

∗

) be the optimal program from x

(or y

∗

f (x

)). Then for any t  2, (x

∗

t+1

, y

∗

t+1

, c

∗

t+1

)

tT

is optimal from x

∗

(or

∗

T +1

= f (x

∗

)). It follows that



∗



= c

∗

for all t ≥ 1,

1.9 Some Applications 65

and, for T ≥ 1,



∗



∞



t=T

t−T



∗



Lemma 9.1 The value function V is increasing, strictly concave on

and differentiable at y

> 0. Moreover,



) = u



(c(y

)). (9.22)

Proof. We shall only prove differentiability of V at y > 0. By concavity

of V on

,atany y

> 0 the right-hand derivative V



) and the left-

hand derivative V



−

) both exist (see Nikaido 1968, Theorem 3.15 (ii),

p. 47) and satisfy



−

) ≥ V



). (9.22a)

Let (x

∗

, y

∗

, c

∗

) be the unique optimal program from y

. Recall that

∗

, y

∗

, c

∗

)isstrictly positive. Take h > 0 and consider the program



, y



, c



) from y

+ h deﬁned as



= x

∗

, c



= c

∗

+ h, y



= y

+ h



= x

∗

, c



= c

∗

, y

∗

= y

for t ≥ 2

Since (x



, y



, c



) is clearly a feasible program from y

+ h, one has

V (y

+ h) ≥

∞



t=1

t−1

u(c



Now, V (y

) ≡



∞

t=1

t−1

u(c

∗

). Hence,

V (y

+ h) − V (y

) ≥ u(c



) − u



∗



= u



∗

+ h



− u



∗



or V



) ≥ u





∗



. (9.22.b)

Take

h > 0, but sufﬁciently small, so that c

∗

−

h > 0. Consider the pro-

gram (x



, y



, c



) from y

− h deﬁned as



= x

∗

, c



= c

∗

−

h; y



= y

∗

−



= x

∗

, c



= c

∗

, y



= y

∗

for t ≥ 2.

66 Dynamical Systems

Since (x



, y



, c



) is clearly a feasible program from y

−

h, one has

V (y

−

h) ≥

∞



t=1

t−1







Then,

V (y

) − V (y

−

h) ≤ u



∗



− u



∗

−



or V



−

) ≤ u





∗



(9.22c)

From (9.22b) and (9.22c)



) ≥ u





∗



≥ V



−

). (9.22d)

But (9.22a) and (9.22d) enable us to conclude that V



−

) = V



) ≡



) = u



∗

Lemma 9.2 The optimal consumption policy function c is increasing,

i.e.,

“y > y



(≥0)” implies “c(y) > c(y



).” (9.23)

Moreover, c is continuous on

Proof. This follows directly from the relation (9.22):



(y) = u



(c(y)) for all y > 0.

Since V is strictly concave, “y > y



> 0” implies “V



(y) < V



).”

Hence

c(y) > c(y



). (9.24)

If y



= 0, c(y



) = 0 and y > 0 implies c(y) > 0.

Continuity of c follows from a standard dynamic programming argu-

ment (see, e.g., Proposition 9.8 and, also Chapter 6, Remark 3.1).

1.9.4.2 On the Optimality of Competitive Programs

The notion of a competitive program has been central to understanding

the role of prices in coordinating economic decisions in a decentralized

economy.

1.9 Some Applications 67

A program (x, y, c) from x

> 0 is intertemporal proﬁt maximizing

(IPM) if there is a positive sequence p = (p

)ofprices such that for

t ≥ 0,

t+1

f (x

) − p

≥ p

t+1

f (x) − p

x for x ≥ 0. (M)

A price sequence p > 0 associated with an IPM program for which (M)

holds is called a sequence of support or Malinvaud prices. A program

(x, y, c) from x

is competitive if there is a positive sequence p = (p

)of

prices such that (M) holds for t ≥ 0, and for t ≥ 1

t−1

u(c

) − p

≥ δ

t−1

u(c) − p

c for c ≥ 0. (G)

A price sequence p > 0 associated with a competitive program (x, y, c)

for which both (M) and (G) hold will be called a sequence of competitive

or Gale prices. The conditions (M) and (G) are referred to as competitive

conditions. One can think of the conditions (G) and (M) in terms of a

separation of the sequence of consumption and production decisions by

means of a price system, the central theme of the masterly exposition of

a static Robinson Crusoe economy by Koopmans (1957). Observe that a

price sequence p = ( p

) associated with a competitive program (x, y, c)

is, in fact, strictly positive. Clearly, if for some t ≥ 1, p

= 0, then (G)

implies

t−1

u(c

) ≥ δ

t−1

u(c) for c ≥ 0

and this contradicts the assumed property that u is increasing. On the

other hand if p

= 0, (M) implies

f (x

∼

) ≥ p

f (x) for all x ≥ 0.

Since p

> 0, we get a contradiction from the assumed property that f

is increasing. Here, we see that p >> 0.

Our interest in competitive programs is partly due to the following

basic result.

Theorem 9.2 If (x, y, c; p) from x

> 0 is a competitive program and

lim

t→∞

= 0, (IF)

then (x, y, c) is the optimal program from x

> 0.

68 Dynamical Systems

Proof. Let (¯x, ¯y, ¯c)beany program from x

> 0. By using the conditions

(G) and (M) one gets



t=1

t−1

[u(

) − u(c

)] ≤



t=1

[

− c

]

T −1



t=0

{[p

t+1

f (

)−p

]−[ p

t+1

f (x

)−p

]}+p

−

)≤ p

Hence,



t=1

t−1

) −



t=1

t−1

u(c

) ≤ p

Now, as T →∞, the terms on the left-hand side have limits, and the

condition (IF) ensures that the right-hand side converges to zero. Hence,

∞



t=1

t−1

) ≤

∞



t=1

t−1

u(c

This establishes the optimality of the program (x, y, c; p).

In his analysis of the problem of decentralization in a dynamic econ-

omy, Koopmans (1957) observed that “by giving sufﬁciently free rein

to our imagination, we can visualize conditions like (G) and (M) as be-

ing satisﬁed through a decentralization of decisions among an inﬁnite

number of agents, each verifying a particular utility maximization (G)

or proﬁt maximization (M) condition. A further decentralization among

many contemporaneous agents within each period can also be visualized

through a more disaggregated model. But even at this level of abstrac-

tion, it is difﬁcult to see how the task of meeting the condition (IF) can be

“pinned down on any particular decision maker. This is a new condition

to which there is no counterpart in the ﬁnite model.” It should be stressed

that if any agent in a particular period is allowed to observe only a ﬁnite

number of prices and quantities (or, perhaps, an inﬁnite subsequence of

∗

), it is not able to check whether the condition (IF) is satisﬁed.

The relationship between competitive and optimal programs can be

explored further; speciﬁcally, it is of considerable interest to know that

the converse of Theorem 9.2 is also true.

We begin by noting the following characterization of competitive

programs.

1.9 Some Applications 69

Lemma 9.3 Let (x, y, c) be a program from x

> 0. There is a strictly

positive price sequence p = ( p

) satisfying (G) and (M) if and

only if

(i) x

> 0, y

> 0, c

> 0 for t ≥ 0

(ii) u



) = δu



t+1

) f



) for t ≥ 0 (RE)

Proof. (Necessity) Using [(U.2] and (G), c

> 0 for t ≥ 1. Since y

≥

,wegety

> 0 for t ≥ 1. Using [F.2], x

> 0 for t ≥ 0. This establishes

(i).

Using (G) and c

> 0, one obtains

= δ

t−1



) for t ≥ 1.

Using (M) and x

> 0, one obtains

t+1



) = p

for t ≥ 0.

Combining the above two equations establishes (ii).

(Sufﬁciency) Let p

= δ

t−1



) for t ≥ 1 and p

= p



). Now,

using (ii), one obtains

t+1



) = p

for t ≥ 1. (9.25)

Given any t ≥ 1 and c ≥ 0, concavity of u leads to

t−1

[u(c) − u(c

)] ≤ δ

t−1



)(c − c

) = p

(c − c

so that (G) follows by transposing terms. Given any t ≥ 0 and x ≥ 0,

concavity of f leads to

t+1

[ f (x) − f (x

)] ≤ p

t+1



)(x − x

) = p

(x − x

using (9.25). Thus (M) follows by transposing terms.

Remark 9.5 The condition (RE), known as the Ramsey–Euler condi-

tion, asserts the equality of the marginal product of input with the in-

tertemporal marginal rate of substitution on the consumption side.

Now we are in a position to establish a converse of Theorem 9.2.

70 Dynamical Systems

Theorem 9.3 Suppose (x, y, c) is the optimal program from x

> 0. Then

there is a strictly positive price sequence p = ( p

) satisfying (G) and (M),

and

lim

t→∞

= 0. (IF)

Proof. Since (x, y, c) is optimal from x

> 0, it is strictly positive i.e.,

(x, y, c) >> 0.

Since (x, y, c) is optimal from x

> 0, for each t ≥ 0, x

must maximize

W (x) ≡ δ

t−1

u(y

− x) + δ

u( f (x) − x

t+1

)

among all x satisfying 0 ≤ x ≤ y

and f (x) ≥ x

t+1

. Since c

> 0 and

t+1

> 0, the maximum is attained at an interior point. Thus, W



) = 0,

which can be written as



) = δu



t+1

) f



Hence, (x, y, c) satisﬁes (i) and (ii) of Lemma 9.3. Consequently, there

is a strictly positive price sequence p = ( p

) satisfying (G) and (M). In

fact, using (G) and c

> 0, we have

= δ

t−1



) for t ≥ 1.

and

= p



Using Lemma 9.1 we also get

= δ

t−1



) for t ≥ 1. (9.25



)

Using the concavity of V and (9.25



), we obtain for any y > 0 and t ≥ 1

t−1

[V (y) − V (y

)] ≤ δ

t−1



)(y − y

)

= p

(y − y

Thus for all t ≥ 1 and any y > 0

t−1

V (y) − p

y ≤ δ

t−1

V (y

) − p

. (9.26)

Choosing y = (y

/2) in (9.26), we get

0 ≤ (1/2) p

≤ δ

t−1

[V (y

) − V (y

/2)] ≤ δ

t−1

V (y

) (9.27)

1.9 Some Applications 71

For any y, V (y) ≤ [b/1 − δ] (see the proof of Theorem 3.1). Also

for t ≥ 1, 0 ≤ x

≤ y

and 0 <δ<1. The condition (IF) follows from

(9.27) as we let t tend to inﬁnity.

Lemma 9.4 The optimal investment policy function i: R

→ R

de-

ﬁned as

i(y) ≡ y − c(y) (9.28)

is increasing, i.e.,

“y > y



(≥0)” implies “i(y) > i(y



).” (9.29)

Moreover, i is continuous on

Proof. Suppose, to the contrary that i(y) ≤ i(y



), where y > y



> 0.

This implies



(i(y)) ≥ f



(i(y



)) (9.29



)

The Ramsey-Euler condition (RE) gives us

δu



[g( f (i(y)))] f



(i(y)) = u



(c(y))

δu



[g( f (i(y



)))] f



(i(y



)) = u



(c(y



)).

Using (9.23) and [U.2], we get

δu



[g( f (i(y)))] f



(i(y)) = u



(c(y))

< u



(c(y



)) = δu



[c( f (i(y



)))] f



(i(y



)).

Hence, by using (9.29



) and [U.2],



[c( f (i(y)))] < u



[c( f (i(y



)))], which leads to

i(y) > i(y



a contradiction. If y



= 0, i (y



) = 0 and y > 0 implies i(y) > 0. Conti-

nuity of i follows from the continuity of c.

1.9.4.3 Stationary Optimal Programs

In view of the assumption [F.4], we see that the equation

δ f



(x) = 1,

(where 0 <δ<1) has a unique solution, denoted by x

∗

> 0. Write y

∗

f (x

∗

) and c

∗

= y

∗

− x

∗

. It is not difﬁcult to verify that c

∗

> 0. The triplet

72 Dynamical Systems

∗

, y

∗

, c

∗

) is referred to as the (δ-modiﬁed) golden rule input, stock , and

consumption, respectively. Deﬁne the stationary program (x

∗

, y

∗

, c

∗

)

from x

∗

> 0 by (again x

= x

∗

)

∗

= x

∗

, y

∗

= f



∗



, c

∗

= c

∗

for t ≥ 1

Deﬁne a price system:

∗



∗



by p

∗





∗



, p

∗

= δ

t−1





∗



for t ≥ 1.

It is an exercise to show that p

∗

= ( p

∗

) is a strictly positive sequence of

competitive prices, relative to which the stationary program (x

∗

, y

∗

, c

∗

)

satisﬁes the conditions (G) and (M). Also, with 0 <δ<1, the condition

(IF) is satisﬁed (since lim

t→∞

∗

= 0). Hence, the stationary program

∗

, y

∗

, c

∗

)

is optimal (among all programs) from x

∗

> 0.

1.9.4.4 Turnpike Properties: Long-Run Stability

Let 0 < x

< x

∗

, and suppose that (x

∗

, y

∗

, c

∗

) is the optimal program

from x

. Again, recall that the sequences (x

∗

, y

∗

, c

∗

)

t≥1

are also optimal

programs from y

∗

Since (x

∗

, y

∗

, c

∗

)

is the optimal program from x

∗

, it follows that

∗

< f (x

∗

). Hence, x

∗

= i(y

∗

) < i( f (x

∗

)) = x

∗

, and this leads to y

∗

f (x

∗

) < f (x

∗

). Thus,

∗

< f



∗



for all t ≥ 1

∗

< x

∗

for all t ≥ 1

∗

< c

∗

= f



∗



− x

∗

for all t ≥ 1. (9.30)

By the Ramsey-Euler condition

δu





∗







∗



= u





∗



Since δ f



∗

) > 1, it follows that u



∗

) < u



∗

). Hence c

∗

> c

∗

. Since

c is increasing, y

∗

> y

∗

. Repeating the argument, y

∗

t+1

> y

∗

for all

t ≥ 1. This leads to x

∗

t+1

> x

∗

for all t ≥ 0, and c

∗

t+1

> c

∗

for all t ≥ 1.

Using (9.30), we conclude that as t tends to inﬁnity, the sequences

∗

), (y

∗

t+1

), (c

∗

t+1

) are all convergent to some

c, respectively.

Clearly, 0 <

 x

∗

,0<

y  f (x

∗

), 0 <

c  c

∗

. Again, using the

Ramsey-Euler condition,

δu





∗

t+1





∗



= u





∗

