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

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1.9 Some Applications 93

Combining (9.57.4) and (9.57.5),

θu((1/2), 1) +θδu(1, 0) = θ V (1/2). (9.57.6)

Using (9.57.6) in (9.57.3), we ﬁnally get

u[(1/2),θ] + δV (θ ) >θV (1/2) + δV (θ )

= θ V (1/2) + δV (2θ(1/2) + (1 − 2θ)(0))

≥ θ V (1/2) + 2δθV (1/2)

= V (1/2),

a contradiction to the optimality equation. This establishes the restriction

(R). Note that this result, and the proof, remain valid if the optimal

transition function h satisﬁes the equation of the tent map:

h(x) =



2x for 0 ≤ x ≤ 1/2

2 − 2x for 1/2 ≤ x ≤ 1

Example 9.4 (The two-sector optimal growth model) The two-sector

model of optimal economic growth, originally developed by Uzawa

(1964) and Srinivasan (1964), is a generalization of the one-sector model,

discussed in Section 1.9.4.1. The restriction imposed in the one-sector

model, that consumption can be traded off against investment on a

one-to-one basis, is relaxed in the two-sector model, and it is princi-

pally this aspect that makes the two-sector model a considerably richer

framework in studying economic growth problems than its one-sector

predecessor.

Production uses two inputs, capital and labor. Given the total amounts

of capital and labor available to the economy, the inputs are allocated

to two “sectors” of production, the consumption good sector and the

investment good sector. Output of the former sector is consumed (and

cannot be used for investment purposes), and output of the latter sec-

tor is used to augment the capital stock of the economy (and cannot be

consumed). Thus, the consumption–investment decision amounts to a

decision regarding allocation of capital and labor between the two pro-

duction sectors.

The discounted sum of outputs of the consumption good sector is to

be maximized to arrive at the appropriate sectoral allocation of inputs in

each period. (A welfare function on the output of the consumption good

sector can be used, instead of the output of the consumption good sector

94 Dynamical Systems

itself, in the objective function, but it is usual to assimilate this welfare

function, when it is increasing and concave, in the “production function”

of the consumption good sector.)

Formally, the model is speciﬁed by (F, G,ρ,δ), where

(a) the production function in the consumption good sector, F :

→

, satisﬁes the following:

[F.1] F is continuous and homogeneous of degree 1 on

[F.2] F is nondecreasing on

[F.3] F is concave on

(b) the production function in the investment good sector, G :

→

, satisﬁes the following:

[G.1] G is continuous and homogeneous of degree 1 on

[G.2] G is nondecreasing on

[G.3] G is concave on

[G.4] lim

K →∞

[G(K , 1)/K ] = 0.

(d) the discount factor δ satisﬁes 0 <δ<1.

The optimal growth problem can be written as

(P)











maximize



∞

t=0

t+1

subject to c

t+1

= F(k

, n

) for t ∈

t+1

= G(x

− k

, 1 − n

) + (1 − ρ)x

for t ∈ Z

0 ≤ k

≤ x

, 0 ≤ n

≤ 1, for t ∈ Z

= x > 0.

Here x

is the total capital available at date t, which is allocated be-

tween the consumption good sector (k

) and the investment good sector

− k

). Labor is exogenously available at a constant amount (normal-

ized to unity), which is allocated between the consumption good sector

) and the investment good sector (1 − n

). Note that an exogenously

growing labor force (at a constant growth rate) can be accommodated

easily by interpreting k

and x

as per worker capital stocks, and reinter-

preting the depreciation factor, ρ.

As in the one-sector model, we can ﬁnd B > 0, such that for x ∈ [0, B],

we have G(x, 1) + (1 − ρ)x in [0, B], and furthermore for any solution

to (P), x

∈ [0, B] for some t. Thus, it is appropriate to deﬁne S = [0, B]

as the state space.

1.9 Some Applications 95

To convert the problem to its reduced form, we can deﬁne the transition

possibility set  as

={(x, z)in

: z ≤ G(x, 1) + (1 − ρ)x}

and the utility function u as

u(x, z) = maximize F(k, n)

subject to 0 ≤ k ≤ x, 0 ≤ n ≤ 1

z ≤ G(x − k, 1 − n) + (1 − d)x.

Then a solution to (P) corresponds exactly to an optimal program for

(, u,δ), and vice versa.

1.9.7 Dynamic Games

Dynamical systems arise naturally from dynamic games that capture

problems of intertemporal optimization faced by an agent (“player”)

who has a conﬂict of interest with other agents.

Think of a renewable resource that grows according to a biological law

described by a function f :

→ R

satisfying the following proper-

ties:

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

[F.2] f is continuous and increasing;

[F.3] there is

y > 0 such that f (y) < y for all y >

y and f (y) > y

for 0 < y <

Thus, if the stock of the resource in period t is y

and no extraction

(harvesting/consumption) takes place, the stock in the next period is

given by

t+1

= f (y

There are two players involved in extraction with identical preferences.

The (common) utility function u :

→ R

satisﬁes:

[U.1] u is continuous, increasing, and strictly concave;

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

lim

c↓0



The (common) discount factor δ satisﬁes 0 <δ<1.

96 Dynamical Systems

Let K = max[y

y] and S = [0, K ]. We refer to S as the state space

and interpret y as the stock at the beginning of each period. In the Cournot

tradition, each player chooses a consumption sequence over time that

maximizes the discounted sum of utilities, taking the actions of the other

player as given. The structure of the model makes it convenient to state

the dynamic optimization in a generalized game form in the sense of

Debreu (1952). (Here we are sidestepping some ﬁner points on model-

ing independent choices with conﬂicting interests, and suggest that the

reader should carefully study the more complete elaboration of Sundaram

(1989).)

Let h

={y

, (c

1,0

, c

2,0

),...,y

t−1

, (c

1,t−1

, c

2,t−1

), y

} be a history of

the game up to period t; the stocks y

and the consumption extraction

choices (actions) up to period t. h

is the set of all possible histories up to

period t.Astrategy g

(i)

= (g

(i)

) for player i is a sequence of functions g

(i)

where g

(i)

speciﬁes the action c

(consumption/extraction) in period t as

a (Borel measurable) function of h

(subject to the feasibility constraint

faced by i at h

, to be formally spelled out below). A (Borel measurable)

function

(i)

: S → S (satisfying g(y) ≤ y) deﬁnes a stationary strategy

(i)

= (

(i)

(∞)

) for player i: irrespective of the history up to period t, and

independently of t, it speciﬁes a consumption c =

(i)

(y) (satisfying

 y). We shall refer to

as a policy function, and often with an abuse

of notation, denote (

(i)

(∞)

) simply by

(i)

Now, think of the dynamic optimization problem of player i when

player j = i [i, j = 1, 2] uses a stationary strategy deﬁned by a policy

function

: S → S, assumed to satisfy 0 ≤

(y) ≤ y for y ∈ S.For

simplicity of notation we write g ≡

. Given the initial y

∈ S, the

problem of player i can be written as

maximize

}

∞



t=0

u(c

i,t

) (9.58)

subject to

t+1

= f [y

− g(y

) − c

i,t

] for t ≥ 0, (9.59)

i,t

∈ [0, y

− g(y

)] for t ≥ 0. (9.60)

Equations (9.58)–(9.60) deﬁne an SDP for player i. Let G

be the set

of all strategies (including history dependent strategies) for player i that

1.9 Some Applications 97

satisfy the constraints (9.59) and (9.60) for each t.Forg

(i)

∈ G

denote

by W

(i)

) the payoff to player i from employing strategy g

(i)

given g,

and y

= y. A strategy g

∗(i)

∈ G is said to be optimal if W

∗(i)

)(y) ≥

(i)

)(y) for all y > 0 and all g

(i)

∈ G

. If an optimal strategy g

∗(i)

exists, the corresponding payoff function W

∗(i)

): S → R is called

the value function of player i, arising from responding optimally to the

policy function (more precisely, the stationary strategy deﬁned by the

policy function) g. Note that if g

∗(i)

and

∗(i)

are both optimal strategies

(given (g

(∞)

) for j ) then W

∗(i)

] = W

[

∗(i)

The following result is derived from a suitable adaptation of results in

Maitra (1968) (see also Chapter 6).

Lemma 9.6 Suppose g : S → S is a lower semicontinuous function sat-

isfying

(i) 0 ≤ g(y) ≤ y for y ∈ S and

(ii) [(g(y) − g(y



))/y − y



] ≤ 1 for y, y



∈ S, y = y



Then

(a) there exists an optimal strategy (

∗(i)

(∞)

) for player i that is stationary

(the policy function being

∗(i)

(b) the value function W

(

∗(i)

) is a nondecreasing, upper semi-

continuous function on S.

For notational simplicity write W

(

∗(i)

) ≡ V

. It should be stressed

that Lemma 9.5 identiﬁes a class of stationary strategies of player j

(deﬁned by policy function g) such that the best response of player i is

also a stationary strategy (optimal in the class of all strategies G

A symmetric Nash equilibrium in stationary strategies is a pair of

policy functions (

∗(1)

∗(2)

) such that

(a)

∗(1)

∗(2)

= g

∗

; and

(b) given

∗( j)

( j = 1, 2),

∗(i)

(i = j) solves the problem (9.58)–

(9.60).

Going back to the optimization problem (9.58)–(9.60), we note that a

symmetric Nash equilibrium results when player i, accepting the strategy

of j deﬁned by a policy function g, obtains g itself as a possible response,

i.e., a solution to (9.58)–(9.60). A symmetric Nash equilibrium (g

∗

, g

∗

)

is interior if 0 < g

∗

(y) < y.

98 Dynamical Systems

One of the fundamental results in the context of the resource allocation

game is the following.

Theorem 9.5 There is a symmetric Nash equilibrium in stationary

strategies (g

∗

, g

∗

). The policy function g

∗

can be chosen to be lower

semi-continuous, and for distinct y





in S,

∗



) − g

∗



)



− y



≤

For a proof of this theorem, the reader is referred to Sundaram

(1989).

Remark 9.8 The symmetric Nash equilibrium in stationary strategies is

subgame perfect (Dutta 1999, Chapter 13). Consider the function

ψ(y) = y − g(y) − g(y) ≡ y − 2g(y),

ψ is the savings function of the entire economy after the consumption of

both players. Then, for y



> y



ψ(y



) −

ψ(y



) = (y



− y



) − 2[g(y



) − g(y



)

ψ(y

) −

ψ(y

)

− y

= 1 − 2

[g(y



) − g(y

)

− y

≥ 0

ψ(y

) −

ψ(y

) ≥ y

− y

> 0.

It follows that in the equilibrium proved above, the generated sequ-

ence of the stock levels (y

) is monotone (see Complements and De-

tails).

1.9.8 Intertemporal Equilibrium

Consider an inﬁnite horizon economy with only one producible good

that can be either consumed or used as an input along with labor (a

nonproducible, “primary” factor of production) to produce more of itself.

Net output at the “end” of period t, Y

, is determined by the production

function F(K

, L

) where K

≥ 0 is the quantity of producible good

1.9 Some Applications 99

used as input (called “capital”) and L

> 0 is the quantity of labor, both

available at the “beginning” of period t ≥ 0:

= F(K

, L

). (9.61)

Here we assume that F is homogeneous of degree 1. We shall switch to

the “per capita” representation for L

> 0. Write y

≡

and k

≡

(when L

> 0) and using homogeneity, we get

≡

F(K

, L

)

= F



, 1



≡ f (k

We shall also assume the Uzawa-Inada condition:

lim

k→0



(k) =∞ and lim

k→∞



(k) = 0. (9.61



)

Individuals in the economy live for two periods: in the ﬁrst period

each “supplies” one unit of labor and is “retired” in the second period.

The total endowment or supply of labor L

in period t is determined

exogenously

= L

(1 + η)

,η≥ 0, L

> 0. (9.61



)

We assume that when used as an input in the production process, capital

does not depreciate. At the end of the production process in period t, the

total stock of the producible commodity is K

+ Y

and is available for

consumption C

(0) in period t. The amount not consumed (saved) is

available as the capital stock in period t + 1

t+1

= (K

+ Y

) − C

≥ 0. (9.62)

The division of the producible good between consumption and saving

is determined through a market process that we shall now describe.

A typical individual born in period t supplies the unit of labor in period

t for which he receives a wage w

(>0). Equilibrium in the labor market

must satisfy the condition

= F

, L

). (9.63)

The individual consumes a part c

(1)

of the wage income in the ﬁrst (work-

ing) period of his life and supplies the difference

≡ w

− c

(1)

100 Dynamical Systems

as capital (“lends” his saving to the capital market). Thus, S

, the total

saving of the economy in period t, is simply

= L

, (9.64)

and this is available as K

t+1

, the capital stock to be used as input in period

t + 1

t+1

= S

. (9.65)

In the second period of his life the retired individual consumes c

(2)

t+1

(1 + r

t+1

, where 1 + r

t+1

is the return to capital (r

t+1

is the ‘interest’

rate). Equilibrium in capital market requires

t+1

= F

t+1

, L

t+1

). (9.66)

Thus, in period t the aggregate consumption C

can be written as

= [w

− s

] + (1 +r

t−1

= F

− K

t+1

+ [1 + F

= K

+ F

− K

t+1

Since F is homogeneous of degree 1, Y ≡ F

K + F

L by Euler’s the-

orem, so that

= K

+ Y

− K

t+1

and we get back relation (9.62).

The individual in period t has a utility function u(c

(1)

, c

(2)

t+1

) =

(1)

)

(2)

t+1

)

1−ρ

, where 0 <ρ<1.

Its problem is to maximize (for any w

> 0)



(1)

, c

(2)

t+1



= u(w

− s

, (1 +r

t+1

)

subject to

0 ≤ s

≤ w

The solution

to this maximization problem is easily found:

≡ (1 − ρ)w

. (9.67)

1.9 Some Applications 101

Thus, the evolution of the economy in equilibrium is described by the

following (using (9.61



), (9.64), (9.65), (9.67) for the ﬁrst and (9.63) for

the second relation):

t+1

= [(1 − ρ)w

]/1 + η

= f (k

) − k



). (9.68)

Combining these we have

t+1

1 − ρ

1 + η

[ f (k

) − k



)]

= α(k

Note that α



(k) =−kf



(k) > 0. The Uzawa-Inada condition on f



(see (9.61



)) is not sufﬁcient to derive the condition (PI) in Example 5.1.

1.9.9 Chaos in Cobb–Douglas Economies

A description of markets in which prices adjust when the forces of de-

mand and supply are not balanced was provided by Walras, and was

formally studied by Samuelson (1947). Subsequently the analysis of a

Walras–Samuelson process challenged the very best researchers in eco-

nomic theory, and some conceptual as well as analytical difﬁculties in-

volving the model were exposed. Here we give an example of chaotic

behavior of such a price adjustment process.

Consider an exchange economy with two (price taking) agents and

two commodities. Denote the two agents by i = 1, 2 and the goods by

x and y. Agent 1 has an initial endowment w

= (4, 0) while agent 2

has w

= (0, 2). Let α,β be two real numbers in the interval (0, 1). The

utility function of agent 1 is u

(x, y) = x

1−α

and that of agent 2 is

(x, y) = x

1−β

for (x, y) in the consumption space R

; a typical

economy e can thus be described by the pair (α, β) which refers to the

exponents of the utility functions of the agents. We shall denote the

space of Cobb–Douglas economies deﬁned in this manner by E

≡

(0, 1) × (0, 1). Of particular interest for our purposes is the economy

e = (3/4, 1/2).

Fixing the price of good y as 1, and that of x as p > 0, the excess

demand function for good x is denoted by z( p; e) and is given by

z(p; e) = 2β/p − 4(1 − α), e = (α, β). (9.69)

102 Dynamical Systems

This is derived from the optimization problem solved by each agent

independently, taking the price p (rate of exchange between x and y)as

given. Each agent i maximizes its utility u

subject to ‘budget constraint’

determined by p. More explicitly, the income m

of agent 1 at prices ( p, 1)

is the value of its endowment w

= 4 p.

The income m

of agent 2 at prices ( p, 1) is the value of its endowment

= 2.

Then the optimization problem of agent i(= 1, 2) is given by

maximize u

(x, y)

subject to px + y ≤ m

x ≥ 0, y ≥ 0. (Pi)

The solution (

) to (Pi) gives us the demand for x and y, coming

from agent i. The excess demand for commodity x in the economy at

(1, p) is deﬁned as

z(p; e) = (

) − 4

(4 being the total endowment or supply of x), and can be easily veriﬁed

as that in (9.69).

For every e ∈ E

there is a unique competitive equilibrium p

∗

(e)

given by

∗

(e) = β/[2(1 − α)]. (9.70)

In particular, for

e = (3/4, 1/2), p

∗

(

e) = 1.

The Walras–Samuelson (“tatonnement”) price adjustment process is

deﬁned as

t+1

= T ( p

, e,λ) (9.71)

where

T (p

, e,λ) = p

+ λz(p

; e), (9.71



)

and λ>0 is a “speed of adjustment” parameter. Of particular interest

for our purpose is the value

λ = 361/100.