Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications

58

C.

De Campos et al.

Previous numerical simulations

of

mixing between basalt and granite,

in the same proportions as in this work (95% granite

+ 5% basalt) and

the same viscosity contrast, point towards mixing probabilities around

zero (Jellinek et aI., 1999). The combination

of

convection and chaotic

dynamics, however, greatly enhances the diffusive processes in molten

silicate melts over filament or lamellae dimensions.

Preliminary analytical results reveal different mobility and mixing

efficiency for different elements, which are presented in this work

as

oxides. This way, despite very low diffusion coefficients and a very

short mixing time (154 min),

CaO and MgO may reach mixing

efficiency coefficients around 5% while

K

2

0 and Na20 come closer to

50% (0.43 and 0.48).

The application

of

this device under geologically relevant conditions

has demonstrated that chaotic mixing processes are very efficient ways

to mix silicate melts.

Acknowledgements: Thanks

to

Philippe Courtialfor REE-rich glasses used

in

sample preparation and INGV /Unrest / Italy forfunding.

References

[1]

K.Righter and

MJ.

Drake. Effect

of

water ... A high pressure and

temperature terrestrial magma ocean and core formation,

Earth Planet. Sci.

Lett. 171:383-399,1999.

[2]

C.C.Reese and V.S.Solomatov. Fluid Dynamics

of

local martian magma

ocean.

/caras. 154: 102-120, 2006.

[3]

Y.Abe. Chemical and thermal evolution

of

the terrestrial magma ocean.

Physics

of

Earth and Plan. Int., 100:29-39, 1997.

[4]

1.M.Ottino. The kinematics

of

mixing: stretching, chaos, and transport.

Cambridge Texts in Applied Mathematics, 1989.

[5]

H.

Aref. Stirring by chaotic advection. 1. Fluid Mech.,

143:

1,

1984.

[6]

1.M.Ottino. Mixing, chaotic advection and

turbulkence.Annu.Rev.Fluid.Mech. 22: 207 -53, 1990.

[7]

M.

Funakoshi. Chaotic mixing and mixing efficiency in short time. Fluid

Dyn. Res.,40:1-33, 2008.

[8]

P.D.Swanson and 1.M. Ottino. A comparative computational and

experimental study

of

chaotic mixng

of

viscous fluids. 1. Fluid Mech., 213

227-249,1990.

[9]

D. Giordanno, 1.K. Russel and D.B. Dingwell. Viscosity

of

magmatic

liquids: a model.

Earth Planet. Sci. Lett. 271: 123-134, 2008)

[10] A.M. 1eIlinek, R.C. Kerr and R.W. Griffiths. Mixing and compositional

stratification produced by natural convection.

1.

Experiments and their

application to

earth's core and mantle.

1.

Geoph. Res. 104 (B4): 7183-7201,

1999.

Multiple

Equilibria

and

Endogenous

Cycles

in

a

Non-Linear

Harrodian

Growth

Model

Pasquale

Commendatore

1

,

Elisabetta

Michetti

2

,

and

Antonio

Pint0

3

1

Dept.

of

Economic

Theory

and

Applications

University

of

Naples

Federico

II

Naples,

Italy

(e-mail:

commenda@unina.it)

2

Dept.

of

Economic

and

Financial

Institutions

University

of

Macerata

,

Italy

(e-mail:

michetti@unimc.i

t)

3

Dept.

of

Statistical

Sciences

University

of

Naples

Federico

II

Naples,

Italy

(e-mail:

antpinto@unina.it

)

59

Abstract:

The

standard

result

of

Harrod's

growth

model

is

that,

becaus

e

investors

react

mor

e

strongly

than

savers

to

a

chang

e in income,

the

long

run

equilibrium

of

the

economy

is

unstable.

We

re-interpr

et

the

Harrodian

instability

puzzle

as a local

instability

problem

and

integrate

his

mod

el

with

a

nonlin

ea

r

investm

ent

function.

Multiple

equilibria

and

different

types

of

complex

behaviour

emerge. Moreover,

even

in

the

presence

of

locally

unstable

equilibria, for a large

set

of

initial

conditions

the

time

path

of

the

economy

is

not

diverging,

providing

a

solution

to

th

e

instability

puzzle.

Keywords:

economic

growth,

Harrodian

instability

puzzle,

capacity

utilisation,

multiple

equilibria,

chaos

.

1

Introduction

In

the

Keynesian

theor

y of growth,

as

developed by Harrod[2]

and

[3],

firms

investment

decisions

can

be

separated

into

two components:

the

first compo-

nent

corresponds

to

the

warranted rate

of

growth, defined as

the

rate

of

capital

accumulation

that

the

economy

can

sustain

in

the

long

run

with

the

saving

corresponding

to

the

e

xp

ecte

d

or

normal level

of

production

or

income;

the

other

component

reacts

to

the

short-run

discrepancies between

current

and

normal

degree of

capacity

utilisation.

Th

e

standard

Harrodian

result, follow-

ing

the

hypothesis

that

investors

react

more intensely

than

savers

to

a change

in income, is

the

local

instability

of

the

equilibrium

rate

of growth.

Most

lit-

erature

has

interpreted

this

result

as

the

outcome

of a

dynamic

analysis

of

global stability. However,

as

argued

by

Sen[6

],

Harrod's instability analysis

over-stresses a local problem near the equilibrium without carrying the stor'y

far

enough. Moreover,

Harrod

himself

[2]

and

[3]

has

stated

that

his analysis

60

P.

Commendatore.

E.

Michelti.

and

A. Pinto

does

not

give a

complete

account

of

the

problem, suggesting

that

he

had

only

tried

to

underline

the

existence

of

some centrifugal forces coming

into

play

as soon

as

the

economy gets

out

of

equilibrium.

The

reference

to

these

forces

does

not

exclude

the

existence

of

other

forces

producing

stabilising effects.

In

what

follows we

present

a

Harrodian

growth

model

integrated

by

a

non

linear

investm

e

nt

function

as

in

Kaldor's

[4]

trade

cycle model.

There

can

be

either

one

or

three

equilibrium solutions.

The

first

corresponds

to

the

warranted

rate

of

growth

while

the

other

two

correspond

to

under

and

over

utilisation.

If

the

initial

capacity

utilisation

is sufficiently close

to

normal,

two different cases

may

occur: a)

the

economy converges

towards

one

of

the

three

equibria; b)

the

economy fluctuates permanently.

Even

in

this

case

the

long

term

evolution

of

th

e

system

is

bounded,

providing a solution

to

the

Harrodian

instability

puzzle.

2

The

model

The

model describes a single-good closed economy.

Production

involves fixed

proportions

of

labour

and

capital

(Leontief technology). Technical progress is

excluded.

The

labour

supply

is perfectly elastic

and

capital

does

not

depreci-

ate. Hence,

at

time

t , we have yt = bL

t

:;

y";;P

=

aK

t

,

being

a,

b > 0, where

y";;P

is

the

potential

output,

yt

the

current

output,

K

t

the

stock

of

capital,

L

t

the

amount

of

labour

employed in

production

all

evaluated

at

tim

e t (no

production

lag).

Parameters

a

and

b are

the

reciprocal of

the

capital

and

labour

coefficients respectively.

In

each

period

the

capital

stock

is

not

fully

utilized, so

that

the

potential

and

current

outputs

do

not

coincide. We define

the

degree

of

capacity

utilisation

as

the

ratio

between

current

and

potential

output:

Ut

=

yt

/

Y";;p.

Saving

corresponds

to

a

constant

fraction of

the

income

of

the

economy:

(1)

where

St

is

the

ratio

saving

to

capital

and

s is

the

(constant)

propensity

to

save

out

of

income.

Investment

is

determined

as

follows:

(2)

The

component

G

in

equation

(2) is

interpreted

as

the

Harrodian

warranted

rate

of

growth

that

corresponds

to

the

saving

generated

at

normal

capacity

utilisation:

G =

sail.

(3)

The

value

il

represents

the

normal

degree of

capacity

utilisation,

that

is,

the

optimal

degree

of

capacity

utilization

given

the

existing technology.

Capital

accumulation

G

t

also

depends

nonlinearly

on

the

current

degree

of

capacity

utilization

Ut.

On

the

nonlinear

component

¢(

Ut)

of

the

investment function

we

make

th

e following

assumption,

which assigns

to

it

a sigmoidal

shap

e (see

Kaldor[4J;

and

Bischi et al.[l]):

Multiple Equilibria and Endogenous Cycles

61

Assumption

1.

Function ¢ is such that ¢( u) =

O.

Furthermore ¢ is strictly

increasing with respect to

Ut

(i.

e.

¢'

(Ut)

>

0)

and a u E (0,

1]

exists such that

¢//(Ut)

~

«)0

iff

Ut

::;

(>

)u.

Investment

increases

with

the

curr

ent

degree

of

capacity

utilisation. How-

ever,

wh

en

the

degree

of

utilisation

is

substantially

below

the

normal

value,

the

entrepreneurs'

inc

enti

ve for

capital

accumulation

is small;

when

the

de-

gree

of

utilisation

is

substantially

above

the

normal

value, inv

estme

nt

slows

down again,

due

to

the

increasing

installation

costs

and

to

the

increasing

risks

of

e

ntering

other

projects.

In

order

to

guarante

e positive

capital

accu-

mulation,

we also

make

the

following

assumption.

Assumption

2.

We assume G +

¢(

O)

> 0 , that is the long

run

expected

growth

of

demand G is always high enou

gh

to induce positive

investments

even

in

correspondence

of

a low capacity utiK5ation

in

the curre

nt

period.

The

dynamics

of

th

e

sy

stem

is

generated

by

the

variations

in

the

degree

of

capital

utilisation

in

the

face

of

discr

epa

ncies between

demand

and

supply

as

follows:

(4)

where

Ut+l

denotes

the

one-period forwarded value of

the

state

variable

Ut

while

parameter

8 is

th

e

adjustment

speed

of

capacity

utilization.

By

substi-

tuting

equations

(1), (2)

and

(3)

into

(4),

we

obtain

the

general model for

the

evolution

of

the

state

variable

Ut

describing

the

degree

of

capacity

utilisation

:

(5)

3

Fixed

points

and

their

stability

Concerning

the

fixed

points

of

(5)

the

following

proposition

holds.

Proposition

1.

Let

u*

be

a steady state

of

map

(5), then the following re-

lat'ion

holds:

u*

= u +

¢(u*)/sa.

Observ

e

that

the

corresponding

equilibrium inv

est

ment

(and

saving) is

given

by

G* = S* = G + ¢( U *).

It

is easy

to

verify

that

the

following

propo-

sition holds:

Proposition

2.

u*

= u is a fixed point

of

(5) for any choice

of

the parame-

ters.

This

proposition

follows

directly

from

the

asumption

¢(u)

= 0, hence

Proposition

1 is verified for u* =

u.

As a consequence,

the

eq

uilibrium

.

corresponding

to

the

warranted

rate

of

growth

always exists. We need

to

verify

the

existence

of

other

equilibria. Observe

that

map

(5)

can

be expressed

as

the

sum

of

a linear function

and

a sigmoidal function,

that

is,

Ut+l

=

f(ut)

= g(Ut) + 8¢('ut) = (hUt +

m)

+

8¢(ut),

where h =

(1

- 8sa)

and

m = 8sau.

Then

we prove

the

following

proposition

about

th

e

number

of

fixed

points

of

f.

62

P.

Commendatore,

E.

Michetti, and A. Pinto

Proposition

3.

If

1'Cu)

:::;

1 then

u*

= u

is

the unique fixed point

of

f ,

otherwise,

if

l'

(u) > 1 then f has three fixed points

(Ul

< u < u

r

).

Proof. Assume 0

:::;

1'(u)

:::;

1

and

observe

that

!"(Ut)

2:

«)0

iff

Ut

:::;

(>

)u,

being

f"(ud

= eq/'(Ut). Simple geometrical considerations prove

that

u*

= u

is

the

unique fixed

point

of

f. Consider now

the

case in which

l'

(

u)

<

O.

Observe

that

1'(Ut) = h +

eq/(ud,

\lUt

> 0

and

that

eq/(Ut)

:::;

eq/(u), hence

1'(Ut)

:::;

h + eq/(u) = 1'(u),

\:IUt

>

O.

Since 1'(u) < 0

thcn

1'(Ut) < 0,

\:IUt

> 0, i.e. f is

strictly

decreasing

and

consequently is has

at

most

one

fixed

point.

Finally

consider

the

case 1'(u) > 1

and

remember

that

f(O)

2:

0

and

that

f is convex (concave) for U :::;

(>

)u.

Trivially f

must

intersect

the

45 degree line in two

more

points,

namely

Ul

and

U

r

such

that

Ul

< u < U

r

.

Observe

that

when

1'(u)

= 1

the

map

passes from a fixed

point

to

three

fixed

points

via

tangent

bifurcation.

When

the

fixed

point

u*

=

'u

is unique,

we

can

draw

some general conclusions

about

its

stability

and

the

final dy-

namics of

the

system,

as

stated

in

the

following proposition.

Proposition

4.

Assume that

u*

= u is the unique fixed point

of

map (5).

(i)

If

0

:::;

1'(u)

:::;

1 then u

is

globally stable; (ii)

~f

- 1 <

1'(u)

< 0 then

Ul

< u <

U2

exist such that u attracts any trajectory starting from an

i.

c.

Uo

E

(V'1,U2);

(iii)

if

1'(u)

:::;

-1

then the generic trajectory diverges in

modulus.

Proof. (i) Trivial. (ii)

If

1'(u) < 0

the

generic

trajectory

oscillates.

In

order

to

study

the

asymptotic

behaviour

of

Ut,

remember

that

only simple

dynamics

can

be

produced

and

consider

the

second

iterate

of

f,

namely

f2.

Then

f2

is

strictly

increasing,

it

is concave (convex) if

Ut

:::;

(>

u)

and

remember

that

(j2(u))'

= (j'(U))2.

Then,

- 1 < 1'(u) < 0

=}

(j2(u))'

E

(0,1)

and

f2

has

two more fixed

points

Ul

and

U2

such

that

Ul

< U <

U2

that

are

points

of

an

unstable

two-pe

riod

cycle

of

f.

This

cycle

separates

the

basin

of

attraction

ofu,

given by

B(u)

= (Ul,U2), from

the

basin

B(oo) =

(-oo,ud

U

(U2'+OO)

whose

trajectories

diverges in modulus.

(iii)

Otherwise,

1'(

u)

:::;

-1

=}

(j2(u))'

2:

1

and,

given its

shape,

j2

cannot

have

other

fixed points, so

that

every

trajectory

diverges in

modulus.

About

the

case

in

which

1'(u)

>

I,

and

the

map

admits

three

equilibria,

we

can

only

draw

conclusions

about

the

instability

of

U.

Anyway, in

order

to

study

the

local

stability

of

the

other

fixed

points

and

to

consider

the

possibility

of

complex

dynamics

to

be

exhibited, we have

to

specify function

4>.

In

particular

we focus

on

the

case

in

which

4>

is

simmetrc

around

the

e

quilibrium

U, i.e.

4>(-

(Ut

- u)) =

-4>(

Ut

- u). A sigmoidal function enjoying

this

property

is given by:

(6)

Assume

l'

(u)

> 1,

th

en two

more

distinct

equilibria

are

present

, as proved

in

Proposition

3,

namely

Ul

and

Ur,

which lie

symmetrically

around

u i.c.

U

r

-

fl =

'U

-

Ul.

Multiple Equilibria and Endogenous Cycles

63

Fig.L

Two

dimensional bifurcation

diagram

in

parameters'

plane

(

Z,8)

being

(3

1 = 0.08,

(32

= 7.5

and

u = 0.5.

Curves

G

i

,

i = 1, ... ,5

separate

the

plane

into

different regimes as

ana

lytically proved.

4

Local

and

global

bifurcation

analysis

We now present some l

oca

l

and

global bifurcations for

the

map

(5) when

¢(Ut)

is

given by (6). Define Z = sa,

the

map

(5)

becomes:

. F(u.t) =

(1-

eZ)'Ut

+ gZu + epi

arctan(

P2(Ut -

n).

(7)

We focus first

on

th

e case F'

(u)

> 1, which implies P1

P2

> Z.

In

such a case

F(ut)

may

be

non monotonic as

stated

in

the

following proposition.

Proposition

5.

Assum

e

pIfh

> Z and

ez

> 1.

Then

map

F is bimodal.

Proof

F is

not

monotonic iff

F'

changes sign.

In

order

to

have two

distinct

sol

uti

ons for

the

equat

ion

[P2(Ut

-

u)]2(1-

eZ) =

-(1

+

e(p1P2

-

Z»

being

P1

P2

> Z,

it

must

be

gZ >

1.

If

Proposition

5 holds

then

em

and

eM are respectively

the

minimum

and

the

maximum

points

of F , where em < u <

eM

and

eM-n,

=

'tt-e

m

· Observe

that

for a given Z , F is bimodal if

the

parameter

()

is sufficiently high.

Let

P1

P2

> Z so

that

F

admits

thr

ee fixed points,

then

the

curve

()

= l

iZ,

namely G

1

,

separates

the

parameter

plane (Z,

()

into two regions (see Figure

1), For

points

below such a curve F is

str

ictly increasing so

that

only simple

dynamics

can

be exhibited, more precisely, every initial condition belonging

to

B(uI)

=

(-00,

'II)

(B(u

r

)

=

(u,

+00» generates

trajectories

converging

to

UI

(u

r

).

Otherwise, for

points

above G

1

,

F is bimodal

and

more compl

ex

dynamics

can

be

produc

ed, We focus on such a case.

In

the

following discussion we

take

into

account

that,

by simmetry,

the

propertie

s of

the

local dynamics

around

the

fixed

point

'/II also

apply

to

'11

'

"

and,

similarly,

the

results concerning

the

minimum

point

also apply

to

the

maximum

point.

We

now distinguish between two cases.

If

em

:s;

F(c,n),

that

is,

the

minimum

point

is above

the

45 degree line,

then

the

fixed points

UI

and

ttl'

are locally

stable

and

no complex dynamics

64

P.

Commendatore,

E.

Michetti, and A.

Pinto

",

Fig.

2.

Staircase

diagram

of

the

map

F for

/31

=

(l.OS

,

/32

= 7.5, Z = 0.4

and

u = 0.5.

(a)

B = 12.9:

the

attractor

Al

is chaotic;

(b)

B = 13.4:

th

e unique

attractor

A

after

the

contact

bifurcation

(occurring

at

B

c:::'.

13.379).

can

emerg

e.

Condition C

m

= P(c",) defines a curve C

2

in

the

param

ete

rs'

plane

(Z

,

())

such

that

for

points

above such a curve

the

minimum

point

is

below

the

45

degree line (see again

Figure

1).

In

fact a necessary condition

for cycles or complex dynamics

to

be exhibited is

that

C

m

> P(Cm), i.e.

pl(nd

< O. According

to

the

previous consideration, in

what

follows we

assume

em > F(c,m).

If

PI(ud

>

-1,

the

only way

nl

may

lose stability is

via

a flip bifur

ca

tion, hence

the

following proposition trivially hold

s.

Proposition

6.

Let

Ul

be a

fi

xed point

of

F.

Then for each

(Z

,

())

in the

region defined

as

. { - B{

"h[).,

}

J!(Ul)

=

(B,

Z)

E

1R+

x (O,lh,8

2

)

:

(}Z

< 2 + [

,8

( .

~

.

)]2

1 + 2

lil

-

11,

the point

Ul

is locally stable.

From

the

it

is easy

to

de

termine

th

e flip bifurcation curve

C

3

in

the

parameter

plane

(Z

,

B)

(see Figure 1).

Assume

Cm

>

P(c

m

),

then

map

F is a non invertible

Zl

Z3

- Zl

map

(see

Mira

et

al.[5])

and

the

set S = [F(cm) ,P(CM)] is

trapping,

i.

e.

pt(S)

~

S, 'It E Z

+.

As a consequence we focus

our

ana

lysis

on

the

restriction of F

to

the

set

S,

that

is,

we

consider

the

structure

of

the

attr

act

or A

~

S owned

by

P for

any

initia.l condition 110 E

8.

The

main

resul

ts

are

summar

ized in

the

following Proposition.

Proposition

7.

Let C

m

>

P(c

m

).

(i)

If

p

2

(c

m

)

< u then P has two

co

-

exist'ing attractors

Al

<;;;:

[F(Cm),p

2

(c

m

)]

and

AT

'

<;;;:

[F(CM),p

2

(cM)]j (ii) at

F2(c

m

)

=

'ii

, a contact bifurcation occur

Sj

(

iii)

if

p

2

(c

m

)

E (il, F(cM)) then

F h

as

(],

nn

'

iqu

e

oJ

tractor A

~

8;

('tv

) at F2(c

m

) =

F(CJ\.t)

a final

global

bifnTcation

OCcurSj

(v)

if

p

2

(Cm)

>

F(

CM) almost all trajectories diverge.

Consider

Propositon

7.

In

case (i)

map

F

has

two coexisting

attractors

At

and

A,.

having

the

sam

e

structure

.

The

y

may

both

consist

of

fixed points,

Multiple Equilibria and Endogenous Cycles

65

<a)

20

0.5 0.6

z

Fig.

3.

(a)

One-dimensional

bifurcation

diagram

W.r.t.

()

for

{31

= 0.08,

{32

= 7.5,

Z = 0.4, U = 0.5

and

Uo

=

Cm.

(b)

One-dimensional

bifurcation

diagram

w.r.t.

Z

for

{31

= 0.08,

(32

= 7.5,

()

= 15, U = 0.5

and

Uo

=

Cm.

m-period

cycles

or

chaotic

sets.

Denote

by

B(AI)

and

B(Ar)

their

basins

of

attraction,

i.e.

the

set

of

points

Uo

E 8

which

generate

trajectories

converging

to

them.

Then

B(AI)

= [F(c

m

),

u)

while

B(Ar)

=

(u,

F(CM)],

hence

the

unstable

fixed

point

u

separates

tha

basins

of

the

two

attractors.

About

the

structure

of

Ai,

i = l, r,

remember

that

the

fixed

point

Ui,

i =

r,

l,

can

be

locally

stable

(i.e.

Proposition

6 applies)

and

that

the

only

way

it

can

lose

stability

is

trhough

a flip

bifurcation,

where

a

stable

2-period

cycle is

created.

After

this

first

bifurcation,

a

period

doubling

route

to

chaos is

exhibited

(as

F

restricted

to

[F(c

m

),

F2(cm)] is

topologically

conjugate

to

the

standard

logistic

map).

This

case is

presented

in

Figure

2

panel

(a).

In case (ii)

the

critical

point

C

m

is

pre-periodic,

proving

the

existence

of

parameter

values for which

the

map

is chaotic. In fact

no

attracting

cycles

may

exist

since

their

basins

cannot

contain

the

critical

point.

Condition

F2(C

m

)

= u define

the

contact

bifurcation

curve

C

4

in

the

parameters'

plane

(Z,

e)

(see

again

Figure

1).

The

contact

bifurcation

gives rise

to

a

unique

connected

attractor

to

which

almost

alla

trajecories

converge.

This

situation

occurs

in

case

(iii)

and

it

is

depicted

in

Figure

2

panel

(b).

When

F2(c

m

) =

F(C

M

)

(case (iv)) a final global

bifurcation

occurs

(8

is

no

long

invariant).

After

such

a

bifurcation,

the

attractor

A

disappears

and

almost

all

trajectories

diverge.

Observe

that,

being

F2(C

m

)

increasing

w.r.t.

e,

fixing all

the

other

parameters

and

letting

e

increase

the

map

passes

from

having

two

attractors,

to

one

attractor

and,

finally,

to

none

(as

almost

all

trajectories

diverge).

This

fact is

confirmed

in

Figure

3

(a).

Finally

the

bifurcation

diagram

w.r.t.

Z is

presented

in

Figure

3

(b).

Observe

that

complex

dynamics

can

be

exhibited

for

intermediate

values

of

Z.

66

P.

Commendatore,

E.

Michetti, and

A.

Pinto

5

Conclusions

The

analysis

presented

in

this

paper

suggests a

solution

to

the

Harrodian

instability

puzzle. A possible

outcome

of

our

analysis is

that,

when

the

warranted

growth

rate

is locally

unstable,

the

economy, moving away from

the

normal

degree

of

capacity

utilisation,

tends

towards

another

equilibrium

with

capacity

utilisation

above

or

below normal.

The

occurrence

of

a degree

of

capital

utilisation

that

even in

the

long

run

may

differ from its

normal

value

may

be

questioned

on

the

ground

that

forecasts

cannot

be

systematically

wrong. Following

Harrod,

however, one

may

assume

that,

when

th

e

capacity

utilisation

is sufficiently close

to

the

normal

value, investors

may

be

unwilling

to

change

their

expectations.

Moreover,

when

the

economy is

subject

to

fluctuations which

are

difficult

to

predict

(as in

our

model in

the

pr

esence

of

chaotic

patt

erns

and

compl

ex

basins

of

attraction)

the

long-run discr

epa

ncy

between

normal

and

current

degree

of

capacity

utilisation

is justified.

In

this

case,

long-term

statistic

al

properti

es, as long-run averages, become very

important

to

understand

economic

proc

esses.

References

1.G. Bischi, R.Dieci, G.

Rodano

and

E.

Saltari.

Multiple

at

tractors

and

global

bifurcations

in

a

Kaldor-type

busin

ess cycle model. Journal

of

Evolutionary

Economics, 11, 527:554, 2001.

2.R.F.

Harrod.

An

essay

in

dynamic

theory

. The

Economic

Journal, 49(193), 14:33,

1939.

3.R.F.

Harrod.

Towards

a

Dynamic

Economics.

London,

1948,

Macmillan.

4.N.

Kaldor.

A

Model

of

Trade

Cycle.

The

Economic

Journal, 50(197), 79:82, 1940.

5.C. Mira, L.

Gardini,

A.

Barugola

and

J.C

.

Cathala.

Chaotic

Dynami

cs

in

Two-

Dimensional

Noninvertible

Maps.

Singapore,

1996,

World

Scientific.

6.A.K.

Sen.

Growth

Economi

cs.

Harmondsworth,

1970,

Penguin.

67

Hick

Samuelson

Keynes

Dynamic

Economic

Model

with

Discrete

Time

and

Consumer

Sentiment

Loretti

1.

Dobrescu,l

Mihaela

Neamtu2

and

Dumitru

Opri§3

1

Australian

School

of

Business,

School

of

Economics

University

of

New

South

Wales,

Sydney

Australia

(e-mail:

2

Faculty

of

Economics

and

Business

Administration

West

University

of

Timi§oara,

Timisoara,

Romania

(e-mail:

)

3

Faculty

of

Mathematics

and

Informatics

West

University

of

Timi§oara,

Timisoara,

Romania

(e-mail:

)

Abstract:

The

paper

describes

the

Hick

Samuelson

Keynes

dynamical

economic

model

with

discrete

time

and

consumer

sentiment.

We

seek

to

demonstrate

that

consumer

sen-

timent

may

create

fluctuations

in

the

economical

activities.

The

model

possesses a flip

bifurcation

and

a

Neimark-Sacker

bifurcation,

after

which

the

stable

state

is

replaced

by

a (quasi-)

periodic

motion.

Keywords:

consumer

sentiment,

Hick

Samuelson

Keynes

models,

Neimark-Sacker,

flip

bifurcation,

Lyapunov

exponent.

1

Introduction

The

economic

empirical

evidence ([1],

[4],

[8])

suggests

that

consumer

sentiment

influences

household

expenditure

and

thus

confirms

Keynes'

assumption

that

con-

sumer

"attitudes"

and"

animalic

spirit"

may

cause

fluctuations

in

the

economic ac-

tivity.

On

the

other

hand,

Dobrescu

et

all.

([2],

[3])

analyzed

the

bifurcation

aspects

in

a discrete-delay

Kaldor

model

of

business cycle, which

corresponds

to

a

system

of

equations

with

discrete

time

and

delay. Following

these

studies,

we develop a

dynamic

economic

model

in

which

the

agents'

consumption

expenditures

depend

on

their

sentiment.

As

particular

cases,

the

model

contains:

the

Hick-Samulson

([6]),

Puu

([7])

and

Keynes

([9]) models, as well

as

the

models

from

[9].

The

model

possesses a flip

and

Neimark-Sacker

bifurcation,

if

the

autonomous

consumption

is

variable.

The

paper

is

organized

as

follows.

In

Section 2, we

describe

the

dynamic

model

with

discrete

time

using

investment,

consumption,

sentiment

and

saving functions.

For different values

of

the

model

parameters

we

obtain

well-known

dynamic

models.

In

Section

3, we

analyze

the

behavior

of

the

dynamic

system

in

the

fixed

point's

neighborhood

for

the

associated

map.

We

establish

asymptotic

stability

conditions

for

the

flip

and

Neimark-Sacker

bifurcations.

In

both

the

cases

of

flip

and

Neimark-

Sacker

bifurcations,

the

normal

forms

are

described

in Section 4. Using

the

QR

method,

the

algorithm

for

determining

the

Lyapunov

exponents

is

presented

in

Section

5.

Finally,

the

numerical

simulation

is done.

The

analysis

of

the

present

model

proves

its

complexity

and

allows

the

description

of

the

different scenarios

which

depend

on

autonomous

consumption.

Skiadas C.H., Dimotikalis I. (editors) Chaotic Systems: Theory and Applications

Подождите немного. Документ загружается.