Economics of Chaos

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The Economics of Chaos or the Chaos of EconomicsAuthor(s): David KelseySource: Oxford Economic Papers, New Series, Vol. 40, No. 1 (Mar., 1988), pp. 1-31Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2663252.

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2/32

Oxford Economic Papers

40

(1988),

1-31

THE

ECONOMICS OF CHAOS OR

THE

CHAOS OF

ECONOMICS*

By

DAVID KELSEY

Abstract

THIS

PAPER

gives

a

simple

account

of

non-linear

dynamics, focussing

on

cycles

and

chaos. There

is a

survey

of economic models which

involve

chaos.

The

case where a chaotic

system

is

subject

to

exogenous

random

shocks

is discussed.

1.

Introduction

Economics and weather

forecastinghave

a

lot in common. When people

are not

talking

about the weather

they

talk

about the economy.

Both

weather forecasters and

economic

forecasters have a bad name with the

public.

The

similarities

go

further:both

are trying to predict

the outcomes

of

very large systems,

the

components

of which

mutually

nteract

n

complex

ways.

The

output

of

both

systems

has a

seeminglyrandomappearance,

even

though

there

are certain other

regularities (e.g., weather

is hotter in

summer han

winter,

also there is

higher employment).

In recent years some mathematicsknown popularlyas the theory of chaos

has

been

studied. This deals

with

systems

of

equations which are able to

produce

motions

so

complex

that

they appear completely random. It is

thought

that

chaos

might

be

applied

both within

fluid mechanics

(which

is

the

body

of

theoretical

knowledge

on which

weather

forecastingdepends)

and

economic

theory.

Anyone

who has

watched the flow of

liquid will realize

how

complex

fluid

motion

can be.

Suppose

that

water

is

flowing

down

a

straight

channel

with

smooth sides. The motion will

break

into a

series of swirls which

have

a

somewhat random appearance. Despite the fact the initial conditions are

symmetric

both

in

space

and

time the flow will be

symmetric

n neither.

Such

fluid flows are described as

being

turbulent.Turbulence s

caused

by

viscosity

or friction

within

the fluid.

Viscosity

introduces non-linear terms

into

the

equations

of

motion

which

allow

complex

turbulent

solutions

to

be

possible.

As an

example,

the

presence

of

viscosity makes aeroplane flight

possible.

In

the absence

of

viscosity pressures

on

the upper

and lower

sides

of the

wing

would be

equal

and hence there would

be

no

uplift.

Because

of

the random

character

of

turbulent motion it has been

suggested

that the

mathematics

of

chaos can be

applied

in

this area. Similar

*

Financial

support

from the ESRC

post-doctoral ellowship

scheme

is

gratefully

acknow-

ledged.

I

would

like to thank

Colin

Sparrow

whose

excellent lectures

greatly encouragedan

interest

n this

subject.

I

am also

grateful

or

comments

rom

MargaretBray, David Canning,

Partha

Dasgupta, Joy Read,

Peter

Read,

Gene

Savin,

Peter

Sinclairand

two

referees of this

journal.

(C

Oxford

University

Press 1988

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3/32

2

THE ECONOMICS OF

CHAOS OR THE

CHAOS OF ECONOMICS

arguments uggest

that it could be of use to economists. This paper aims

to

give an

introduction o the

applications

of chaos

in

economics.

The models in this paper are very simple.

The reason for this is that a

detailed mathematical heory

has

only been

constructed or

one-dimensional

dynamicalsystems. (That is there is only one dependent variable and the

system

of

equations

is

of

the first

order

in

that

variable.)

In

higher

dimensions

there is

no

general theory. There

have been theories developed

of

particularequations,

two

commonly

studied examples

are

the Lorenz

equations (Sparrow (1982),

Guckenheimer and Holmes

(1983))

and the

Henon map (see

Henon

(1976)). Higher

dimension

systems

have also been

studied by

means

of

computer

simulations.These

systems

can

display

all the

kinds

of

behavior discussed

in this

paper

and

other forms of

complex

behavior

as well. Those

systems

studied to date have

largely

been motivated

by applicationsof dynamicsystems theoryin the physicalsciences.

Thus

economic models

involving

chaos are

not

always particularly

realistic.

The

reason

is,

as

explainedabove,

our

research s as

yet

in

its

early

stages.

The

simple

models studied here are

interesting

for

the

following

reasons. Most important

s

that

they

tell us the

kinds of

behavior,

of

which

an economic system

is

capable. Before

this

literature,

t

was largelyassumed

that

in

the

long run,

an

economy

would

be

in

a

stationary tate (or

balanced

growth). Cycles were considered,

but

were

under-emphasizedgiven

the

considerable amount

of

empirical

evidence to

support the

view that the

economy

is

cyclic. Aperiodic

motion

was

essentially

not

considered.

Economic

data is

clearly aperiodic. This,

however, was put

down to the

superposition

of random shocks onto an

essentially stationary

economic

system.

Non-linearities

in

economic

systems

cannot

be

denied.

Little

attention, however,

was

paid

to

them. Here we

give

an

account

of some

aspects

of

the behavior

of

non-linear

systems.

2.

Some results from non-linear dynamics

2.1.

Linear and non-linear difference equations

The

simplest

difference

equation is linear

and first order:

xt+l

=

axt

(2.1)

to

which the solution

is

xt

=

x0at.

This

grows

exponentially

f

laI

>

1.

When

O

a


4/32

D.

KELSEY

3

Xt+ll

XI

I

Xt

XI

X(

1/2

Xt

1

FIG.

1

first-order,

but

are non-linear

then

the behavior

we find is

very

different.

Equation

(2.2)

like

equation

(2.1)

is a

first-order

difference

equation.

Xt+i=fr(xt)

where

fr(x)=rx(1-x):

O

-r

e

-4.

(2.2)

Equation

(2.2)

is

often known

as the

logistic equation.

While equation (2.2)

is also

of

the

first order it is quadratic,

and hence non-linear.

Quadratic

behavior s the

most

simple

kind of

non-linearity.

Hence one might expect

(2.2)

to behave

in

a fairly simple manner as (2.1)

does. In fact

as we shall

see the

solutions to

(2.2)

can

be extremely

complex.

As shown in Fig. 1, fr

has the

following properties: fr(O)

=

fr(l)

=

0.

fr has

a unique

maximum at

_

1

X

-

A

stationary

solution

of

equation

(2.2) is a sequence

(xt)

which is a

solutionof

(2.2)

and is such that

Vt,

xt

=

x

for some

xc

such that 0

-x c

1.

Graphically

a

stationary

solution

occurs at a

point

where the graph

of

fr

crosses the 450

line

(e.g.,

at

Yr

in

Fig. 2).

It is

easily

seen

that

0

is a

stationary

olution

for

all

values

of

r.

Figure

1

can also be used to illustrate the

dynamics of the

system.

Suppose

we start at xo. The

height

of

f(xo)

=

x1

gives

the second state

of

the

system.

We

can

find

the

point corresponding

o

x1

on

the horizontal

axis by

reflecting

n

the

450

line.

A

similar process of going

up to the function

and

reflecting

n

the

450

line gives the next state of the system x2. Further

points

in

the evolution

of

the

system

can

be

found in

a

similar

manner. The

successive

points

in

the solution,

path

xO,

x1,

x2,...

are

getting

closer and

closer

to

the

origin.

This will

happen

for

any

starting point. In Fig. 1 the

unique

stationary

tate at

the

origin

is

(globally)

stable.

Now

imagine

that

we are

gradually ncreasing

the parameter

r,

and we

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5/32

4 THE ECONOMICS

OF CHAOS OR THE CHAOS OF ECONOMICS

xt?1

XO

xI

x2

Yr

1/2

xt

1

FIG. 2

wish

to observe how

this affects the

properties

of

the solution.

The function

fr

has a

single hump.

As

r is increased the

height

of this

hump

increases.

When

r

is

greater

than 1

the

diagram

will

look like Fig.

2. The graph of fr

crosses the

450

line at two points-the origin and

Yr

=

(r

-

1)/r. In fact for

1

-

r

-

4, these

are

the

only two stationary solutions of the equation. A

change

in the

dynamics

of the

system has occurred. Starting from a typical

point x0, subsequent

values

of

x

no

longer

tend to the origin but instead

converge

on the

stationary point

at

Yr.

For

1

-

r

-

3 the origin is unstable

and

Yr

is

stable.

Indeed

it

can be shown

that from

almost every

initial

value

the

path

will

converge

to

Yr

=

(r

-

1)/r.

The

exceptions

are the

origin

which

is also

a

stationary point,

and

1,

which is

mapped

to

the

stationary point

at

the

origin

(fr(1)

=

0).

Thus if

0

x'

(0)

=

1/zj-'(0)

=

1/0

which

proves (ii).

Now

zj-1(y)

>

0

whenever

y

> 0 hence

using

(A.4)

xAM)

0 which

proves

(iii).

Lemma

A.2: For

all

sufficiently

large

a

we can find

i(a)

?

y*(a)

such that:

(i) U2(M*)>Ul(ft)

or

X(j*)>ft

(ii)

u2(ft)

U1(0p*)

or

X(ft)

0p*

provided

that

e1

+

e2

>

1/e2

and

e2

l/e2

it

follows that

el

+

e2

>

1.

Hence we can

choose

f

such

that 1

-

e2

1/e2

ande2


31/32

30

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CHAOS

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THE

CHAOS OF

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Economics of Chaos