重要混沌经济学课件

Oligopoly

theory

is

one

of

the

oldest

branches

of

mathematical

economics

datedback

to

1838

when

its

basic

model

was

proposed

by

French

economist

CournotI.

The

Cournot

duopoly

Kopel

ModelIn

the

recent

literatures,

it

is

also

demonstrated

that

the

oligopolistic

marketsmay

become

chaotic.Among

the

first

who

have

shown

the

Cournot

model

may

lead

to

complexbehavior,

such

as

periodic

and

chaotic

behavior,

was

Puu

[9][10]

.T.

Puu,

Chaos

in

duopoly

pricing,

Chaos

Solitons

Fractals

1

(1991)

573-581.T.

Puu,

The

chaotic

duopolists

revisited,

J.

Econom,

Behav.

Organ.

33

(1998)

385-394In

this

work,

we

consider

a

general

case

of

a

duopoly

model:The

Cournot

duopoly

Kopel

Model

[11][11]

Kopel,

M.,

”Simplex

and

Complex

Adjustment

Dynamics

in

Cournot

DuopolyModels,”

Chaos,

Solitions

and

Fractals,

7,

2031-2048,1996.1I.

The

Cournot

duopoly

Kopel

Model☺Previous

work:Analyzing

chaotic

behaviornumerically☺Our

work:A

rigorous

proof

for

existence

of

chaos

frommathematicalpoint

of

view

isgiven.Two

different

types

of

intermittent

chaos

in

this

model

are

foundand

analyzed.23混沌经济学Day于1982年将非线性动态引入到经济学中，引发了人们对传统经济学的反思，为人们提供了崭新的视角宏观经济中存在混沌现象在微观经济学领域，厂商或者其他经济个体所经营产品的价格、生产或销售的产品数量都可能产生波动，呈现出混沌动态4.1

一个古诺双寡头经济模型描述X和Y代表两个寡头厂商厂商X和厂商Y在t+1时间段生产的产品数量分别用x(t+1)和y(t+1)表示Nash平衡点：古诺双寡头Kopel经济模型[7][7]

Kopel

M.

Chaos,

Solitons

&

Fractals,

1996,

7:

2031~204844.2

分形分析分形图：平衡点周期混沌平衡点混沌周期，5光滑经济周期非光滑经济周期混沌由光滑经济周期演变为混沌：64.2

分形分析混沌吸引子共存现象两个共存的混沌吸引子7吸引域由于对称性，混沌吸引子共存现象普遍存在4.2

分形分析4.3

混沌吸引子的计算机辅助证明将双寡头Kopel模型改写为向量形式：其中：的研究映射 动态（

：）定义为 以此类推得到：，84.3

混沌吸引子的计算机辅助证明定理Kopel经济模型具有如下性质：关于四边形的映射 存在一个闭的不变集 ，使得与2个符号的移位映射半共轭，且因此，当 时，古诺双寡头Kopel经济模型有正拓扑熵。94.4

间歇混沌特性分析PM-I型间歇混沌：分形图104.4

间歇混沌特性分析分岔前后x的时间序列分岔前，分岔后，过渡混沌116倍周期点4.4

间歇混沌特性分析结论：服从幂指数为-0.496的幂律分布PM-I型间歇混沌：层流态平均持续时间分布幂指数特征值:

-0.512诱发激变导致的间歇混沌：4.4

间歇混沌特性分析分形图发生激变前，发生激变前，发生激变后，发生激变后，134.4

间歇混沌特性分析结论：服从幂指数为-0.65的幂律分布诱发激变导致的间歇混沌：层流态平均持续时间分布幂指数特征值:

[-3/2,-1/2]1415在经济学系统中出现的间歇混沌现象可以解释为系统本身具有调节机制，不借助于任何外力，系统总是能够将混乱的市场调整回

(相对)平稳状态或者解释为系统有记忆机制，总是能够记住混乱前的状态并恢复4.4

间歇混沌特性分析4.5

长期平均利润分析混沌能否带来更多的利润？混沌动态的平均利润:非零平衡点:164.5

长期平均利润分析结论：混沌市场并不是完全有害的174.6

控制混沌到Nash平衡点定理考虑受控的古诺双寡头Kopel经济模型，平衡点

是局部渐近稳定的，当且仅当稳健的经济市场仍然是人们最需要的184.6

控制混沌到Nash平衡点）仿真结果（令19小结：重点研究了一个古诺双寡头经济模型中的各种混沌动态，从理论上证明了混沌存在性，并分析了混沌对利润的影响，得到了混沌并非完全有害的结论四、经济系统中的混沌动态研究20Remarks:Both

firms

must

consider

the

actions

and

reactions

of

thecompetitorThe

competitors

have

choose

their

actionssimultaneouslyEach

firm

forms

the

expectation

on

the

quantity

of

the

other

firm,

whichdepend

on

their

own

quantity

and

the

quantity

of

the

other

firm

bothproduced

in

the

previous

period,

in

order

to

determine

a

profitmaximizingquantity

to

produce

in

the

nextperiod.I.

The

Cournot

duopoly

Kopel

ModelModel

descriptionConsider

two

firms

X

andY:(1)Where, denote

the

goods

quantities

that

firm

X

and

firm

Y

producein

period

t,

respectively.21I.

The

Cournot

duopoly

Kopel

ModelNash-equilibria

of

the

KopelmodelThe

fixed

points

(Nash-equilibrium

)

of

system

(1)

satisfy

theequations:(2)The

solutions

of

Eq.

(2)

give

four

equilibria:ⅰ

ⅱⅲforⅳforRemark:

The

fixed

points

depend

on .

In

case

ⅱ,

we

should

have(positive

solution).Also,

in

case

ⅲ

and

ⅳ,

we

should

have (real

solution).22bifurcation

diagram

provides

a

general

view

of

the

evolution

process

of

the

dynamicalbehaviors

by

plotting

a

state

variable

with

the

abscissa

being

one

parameterI.

The

Cournot

duopoly

Kopel

ModelBifurcation

analysisbifurcation

diagram:

rich

and

complexdynamicsFig.

1

Bifurcation

diagram.

(a)

Fix,and.(b)Fixand(b)(a)23I.

The

Cournot

duopoly

Kopel

ModelObservation

of

chaotic

attractors

and

basins

of

attractionSmooth

CycleLost

ofSmoothnessChaoticFig.

2

Different

attractors

in

Kopel

model.

(a)

One

smooth

invariant

cyclewith,..

(c)

Chaotic

attractor

with(b)

Invariant

cycle

loses

its

smoothnesswhen

,.(a)24(b)(c)I.

The

Cournot

duopoly

Kopel

ModelCoexistence

of

two

chaotic

attractors:Fig.

3

Two

chaotic

attractors

coexist

with

different

initial

conditions

when

.(a)

Phase

portraits

of

the

two

chaotic

attractors

and

the

four

Nash

equilibria;

(b)

The

basins

ofattractions.(a)25(b)I.

The

Cournot

duopoly

Kopel

ModelHorseshoe

Chaos

in

the

modelA

convenient

expression

for

the

Kopel

model

is

described

as

follows:(3)whereAfter

a

great

many

trial

attempts,

we

will

discuss

the

dynamics

of

system

(3)with under

the

map and

obtain

that

there

exists

ahorseshoe

in

this

attractor.26We

take

a

proper

quadrangle

|ABCD|

to

be

a

subset

in

the

plane

with

its

four

verticesbeingI.

The

Cournot

duopoly

Kopel

ModelFig.

4

The

attractor

whenand

the

proper

quadrangle.27Fig.

5.

Thequadrangleand

its

image.I.

The

Cournot

duopoly

Kopel

Modelcorresponding

to

the

quadranglefor

which,

there

is

semi-conjugateTheorem

1.

For

the

mapexists

a

closed

invariant

setto

the

2-shift

map.Hence,.28I.

The

Cournot

duopoly

Kopel

ModelProof.We

select

two

appropriate

subsets

inthe

yellow

quadrangle

in

Fig.

6,

with

andThen

under

the

map

, is

mapped

to.

The

first

one

is

denoted

by

a

as

shownbe

its

left

and

right

edge,

respectively.which

is

on

the

right

side

of

the

edge

,and is

mapped

to which

is

on

the

left

side

of

the

edge.To

prove

the

above

theorem,

we

should

find

two

mutually

disjointed

subsets

ofsuch

that

there

exists

a -connected

family

with

respect

to

them.Fig.

6

Thesubsetand

its

image

under

the

mapwith.29I.

The

Cournot

duopoly

Kopel

Model,The

second

subset

is

the

purple

quadrangle,

denoted

by

b,

as

shown

in

Fig.

7.

Takeand to

be

the

left

and

right

edge

of

b,

respectively.

Then

theimageis

on

the

left

side

of

the

edge and

theimage is

on

the

right

side

oftheedge

.Fig.

7

Thesubsetand

its

image

under

the

mapwith.30I.

The

Cournot

duopoly

Kopel

ModelUpon

the

above

simulation

results,

it

is

easy

to

see

that

the

subset

a

and

b

are

disjointedand

it

follows

that

for

every

connection

v

with

respect

toa

and

b,

theimagesand lie

wholly

across

the

quadrangles ,

that

is

to

say,

theimagesand are

still

connections

with

respect

to

a

and

b.

According

totopological

horseshoe

Theorem,

there

exists

a -connected

family,

which

meansthatis

semi-conjugate

to

the

2-shift

map.

Hence,

based

on

the

Lemma,

we

know

that

th