Discrete Economic Dynamics

7/25/2019 Discrete Economic Dynamics

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Dynamic Macroeconomic Theory

Thomas LuxUniversity of Kiel

Prof. Dr. Thomas Lux, [email protected]


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In this lecture: Focus is mainly a deterministic dynamic system without stochasticshocks

However: Adding small amounts of noise does mostly not change the qualitativeoutcome of the dynamics so that in theoretical purpose the analysis of the deterministicanalogue is appropriate.




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Variants of dynamic equations:

1) difference equation:

equivalently:

(2) differential equation:

but there might be also higher derivatives:

(3) mixed difference differential equations:

higher derivatives, more lags are possible

nonlinear equations:


( )tgyayay ttt +++= ...2211

( ) ( )tgyayay ttyy

t

tt

...1 2211

1

++=

=

( )tyyf tt ,...,, 21 =

( )tyhdt

dyt

t ,=

( ) ( ) ( )tgyayayaya nnnn =++++

1

1

10 ...

( ) ( ) ( ) ( ) ( )tgwtybtybwtyatya =+++ 1010

( ) = nttt yyyg ,...,, 1



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1.1.1. Mathematical Background

First-order linear difference equation:

with time-dependent function this is called a non homogenous equation

Homogeneous equation :

or:

Solution via iteration:


( )tgycyc tt =+ 101

( )tg

0

0

1

101

=+

=+

tt

tt

byy

ycyc

1

0,c

cb=

( ) etc.02

01

0

byby

bAbyy

Ay

=

==

= ( )

( ) Ab

yby

t

t

t

=

=

0

1.1 First and Second Order DifferenceEquation


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is a general solution as it satisfies

irrespective of A

to fix A, any known function value could be used, e.g.

types of dynamic behavior:

monotonic convergence

oscillatory convergence

monotonic divergence

oscillatory divergence

( )Aby tt =

( )

( ) 01

1=+

tt y

tt

y

AbbAb

t*

t

(-b)

y* AAbyty == *)(**)*,(

:1

:1

:01

:10


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Gandolfo,1997. Fig.3.1



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( )tgAbycAbyc tt =+++ 101 )()(

Solution of the non-homogeneous equation:

General principle: Solution of non-homogeneous equation consists of solution of

homogeneous equation plus so-called particular solution

Hence: solution for

Particular solution can often be interpreted as a steady state equilibrium in economicmodels

It can be easily checked: if

is also a solution!

( ) =+ tgycyc tt 101 yAby t

t += )(

( )tgycyc =+ 01



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Determination of particular solutions:

Try a function with the same form of g(t) but with undetermined constants

Substitute into non-homogeneous equation and determine the coefficients

Examples:

General solution:


( )

01

01

)1(

cc

ayacc

yatg

+===+

==

:try

101

0 )(cc

aA

c

cy

t

t ++=



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( )

t

t

ttt

tt

dcdc

Bdy

cdc

BdC

BdCcCdcd

BdCdcCdc

CdydBtg

0101

01

1

101

,

0)(

: try)2(

+=

+=

=+

=+

==



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yAyty +== 00 )0,(

Determination of the constant A:

for example, for:

Knowledge of leads to:


( )....

sincos: trysincos)3( 21 ttytBtBtg +=+=

yAccy

t

t += )(1

0

yyyc

cyyyA

t

t +== )()( 01

00



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Second-Order Linear Difference Equations

General form:

Homogeneous equation:

or:

)(c 20112 tgyycyc ttt =++

2

)4(

0)(

0

2/1

2

2

12,1

2

1

22-t

2-t

2

1-t

1

t

1aaa

aa

aa

=

=++

=++

In analogy to first-order equations: try a function like:t

ty

2

0

2

1

c

c2c

c

12211

20112

a,a,0a0c

===++=++

ttt

ttt

yyayyycyc

characteristic equation

two solutions



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Cases:

(1) real-valued solutions:

Combine both solution into:

two constants, because two initial conditions are needed to solve a second-orderequation

Convergence requires:

(2) identical solutions:

Since one only has one solution, one tests as a second solution


04- 22

1>= aa

ttt AAy 2211 +=

1,1 21


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(3) complex numbers with imaginary part:< 0

Try a solution:

would be real-value if A, A were complex conjugate!

tt

t i(Ai(Ay )) ++=

ty

212

12121 42

11

2

1 /, )aa(-ai ==


S d O d Li Diff E i


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Prof. Dr. Thomas [email protected]

Coordinate transformation to polar coordinates:

modulus or absolute value

we can write:

2122

222

sin,cos

/)(r

rrr

+=

+===

tt

t )ir(rA)ir(rAy sincossincos ++=

Re()

Im()

r


S d O d Li Diff E ti


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Using de Moivres theorem:

assume:

real numbers

Potential for true oscillatory motion already with two lags!period: 2/ , amplitude: depends on r

[ ]

[ ]iAAAAAA

tAtAr

tiAAtAArtitrAt)itrAy

nin)i(

t

t

tt

t

n

)(,

)sin()cos((

)sin()()cos()()sin()(cos()sin()(cos(

)sin()cos(sincos

21

21

=+

+=

++=++=

=

==+

=+=

biAAaAA

ibaAibaA

2)(,2

,


S d O d Li Diff E ti


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since:

explosive > constant oscillations if = 1

dampened 1 can be excluded by the condition: 1,2< 1

0)1(,0)1(

1)1(,1)1( 2121

>>

+=++=

ff

aafaaf

212)( aaf ++=

f()

1-1


Second Order Linear Difference Equations


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Stability Conditions (sufficient and necessary):

Stability can be checked without explicit solution of difference equation!


( )

( ) 011)3(

011)2(

011)1(

21

2221

21

>+=

>++=

aaf

aa

aaf


Second Order Linear Difference Equations


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Solution of non-homogeneous equation: add particular solution like before

example:

y

cccgttt

ttt

AAy

gbcbcbcby

gycycyc

2102211

012

20112

:

++

++=

=++=

=++


Determination of constants

via twoinitial conditions

example: lead to

,

21

22111210

10

,

),1(),0(

AA

yAAyyAAy

ytyt

++=++=

==

1 1 2 Economic Application: Multiplier


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If second-order difference equation potentially generates sinusoidal fluctuations

business cycle explanation

Two lagged adjustment are in principle sufficient to generate economicfluctuation

First model of the business cycle: Samuelson, 1939


1.1.2 Economic Application: MultiplierAccelerator Interaction

:0


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Model structure:

consumption function with time lag, 0 < b< 1(multiplier)

investment

constant part: public expenditures

induce investment: accelerator

goods market equilibrium


ttt

ttt

t

ttt

t-t

IC Y

CCkI

GI

III

bYC

+=

=

=

+=

=

)5(

)()4(

)3(

)2(

)1(

1

1

1.1.2 Economic Application: Multiplier


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0)1(

1

)1(

)1(

)(

)(

""""

2

21

211

11

21

=++

=

++=

=++

++=

=++=

aa

t

t-t-t

t-t-t-

ttt-t

bkkb

b

GY

YbkYkbYYY

GbkYYkbY

GYYkbbY

GCCkbYY

particular solution: try

stability: characteristic equation:


Combine the equations:

1.1.2 Economic Application: Multiplier-


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Stability conditions:o.k.

product of the roots

o.k.

Goods market equilibrium is stable, if

Oscillations:


2

22

)1(

4if0

4)1(

/11

0)1(1

(?)01

010)1(1

k

kb

bkkb

kbkb

bkkb

bk

bbkkb

+

>

>++

1.1 First and Second Order Difference


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Gandolfo,1997. Fig.6.1


Prof. Dr. Thomas Lux, WSP1 Room 507, [email protected], +49 431 880-3661

1.1.2 Economic Application: Multiplier


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[ ]{ }

0)1()1)(1()1(

)1(

0)1()1)(1()1()1(

)1()1()1)(1()1(

)1(

)1(

2

20

20

22

021

0

>+++++

+=

=++++++

+=+++++

+=

+=

t

t

tttt

t

t

gbkgkbg

gGy

gGbkgkbgAg

gGgbkAgkbgA

gAy

gGG

fluctuations around growth path: assume

for particular solution: try

pp pAccelerator Interaction

1.1.3 A First Look at Anticipation:


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One finds that is valid particular solution since

it satisfies the difference equation

Assume g(t) is not a known function, but a sequence of (stochastic) realizations ofexogenous variables

different approach for determination of particular solution: operational method

use lag operator

Application to first-order equation:

pBackward and Forward Solutions

)1(

)1(

1

1

tt

tt

tt-t

xbLy

xybL

xbyy

+=

=+

=+

=

===

0 0

)()(i i

it

i

t

iixbxLby t

1

1

1 ,, +

=== ttnttn

tt yyLyyLyLy



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This holds because:

Operator expansion like Taylor-series expansion about 0!

Note: these sequences converge only if |b| < 1 or || < 1, i.e. if the system is stable

If |b| > 1, consider:

which also fulfills the difference equation and is bounded since |1/b| < 1.

=

=+++=0

221 1)1i

iiL...LLL-(

pBackward and Forward Solutions

it

i

i

bt

i

i

i

bt xxLy +

=

= == 11

1

1

)()(



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Particular solution here = geometrically declining sum of all past or future values of x0depending on whether the equation is stable or unstable

alternative expansion:

Derivation, reformulate and develop the second term in a Taylor series of

Application: forward looking models are typically mathematically unstable, but can berepresented via second type of solution concept for particular solution.

Backward and Forward Solutions

)()11

11

=

=i

ii

LL(

11111 )1()1 =L

LL(



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Illustration: Cobweb model

equilibrium price is particular solution,

since b < 0: improper oscillations

stable if supply should have smaller slope than demand


1

11

1

1

1

111

111

)(

,

bb

aa

b

bAp

b

aap

b

bp

aapbbpSD

pbaSbpaD

t

t

tt

tttt

tttt

+=

=

==

+=+=

11 ++=

=+++

=+++

=+++

==

+

++

xppp

xppp

pppp

tttt

tttt

t

e

tt

e

tin a deterministic context: perfect foresight:

homogeneous part leads to characteristic equation:

Discriminant:

Since a2=1stability conditions are violated; note:one stable, one unstable


1.1.3 A First Look at Anticipation:Backward and Forward Solutions


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Particular solution: Operational method for knownsequence of

for second-order equations:

Denote by F = L-1the forward operator:

with F1, F2roots of the polynomial which coincide with1, 2

)(1

1

1

)1(

))((

21

2

2221

2

21

2

21

2211

21

FFFF

tt

tt

tttt

aFaFLLaLa

XLaLa

y

XyLaLa

Xyayay

=

++=++

++=

=++

=++


1

1

tt xX



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it

i

it

i

i

tLtLt

LLLL

XX

XXy

LL

LLFLLF

FFLLaLa

=

=

+=

+=

=

=+=

=

==

==++

0

221

0

11

11

2

111)1)(1(

1

21

21

212

21

,

)1)(1(

))((

))((1

2

2

1

1

21

2

21

1

2

2

1

1

21

Hence:

and

hence:

backward solution




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or

In the RE cobweb case: backward solution for stable root, forward solution for unstableroot.

particular solution is geometrically weighted average of all past, present and (known)future shocks.

ttt

t

it

i

iit

i

i

it

i

iit

i

it

it

i

iit

i

it

pAAp

XX

XXp

XXy

i

i

++=

Discrete Economic Dynamics

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