11 difference equations

7/27/2019 11 Difference Equations

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DIFFERENCE EQUATIONS



PRELIMINARY

A difference equation relates the value of a

dependent variable in one period to its value in one

or more adjacent periods, as well as to the value of

one or more independent variables.

The difference between the largest and smallest

time period indexing the sequence of dependent

variables is the order of the difference equation.

t t t yax x 1

t t t t ybxax x 21



A solution to the difference equation is a

representation of the entire sequence {xt} as a

function of time itself and of the sequence of the

independent variable.

Initial vs. terminal value

The differences between adjacent terms in the

sequence generated by the difference equation

are of the same (alternating) sign if a > (<) 0. In this

case the dynamics generated by the equation are

monotonic (oscillatory).

PRELIMINARY

t t t yax x 1



Stable vs. unstable equilibrium. The sequence

generated by difference equation

converges to the steady state value

if -1 < a < 1. The sequence diverges if a > 1 or a < -1.

What if a = 1? a = -1?

PRELIMINARY

yax x t t 1

t t ya

x

1

1



SOLUTION TO FIRST-ORDER DIFFERENCE

EQUATION

yax x t t 1

yax x t t 21

y yaxa x t t )( 2 yay xat

2

2

yay ya xa x t t 2

3

3

1

0

0

t

i

it

t ya xa x

ya

a xa x

t t

t

1

10




EQUATION

t t t yax x 1

121 t t t yax x

t t t t y yaxa x )( 12t t t yay xa 12

2

t t t t t yay ya xa x 12

2

3

3

1

0

0

t

i

it

it

t ya xa x

• stable if -1 < a < 1 and {yt} is bounded.

• backward solution




EQUATION

t t t yax x 1

121 t t t yax x

t t t t y yaxa x )( 12t t t yay xa 12

2

t t t t t yay ya xa x 12

2

3

3

1

0

n

i

it

i

nt

n

t ya xa x

• stable if -1 < a < 1 and {yt} is bounded.

• forward solution



The general solution to the difference equation

is the sum of the solution to the homogeneous

equation and the particular solution, i.e.,

At t = 0,


EQUATION

yax x t t 1

.1

1 y

aCa x t

t

ya

Ca x

1

10

0 ya

xC

1

10

y

a

a y

a

x x t

t

1

1

1

10 y

a

a xa

t t

1

10



PHASE DIAGRAM

xt-1

xt

xt = xt--1

3002

11 t t x x

400

600400



PHASE DIAGRAM

xt-1

xt

xt = xt-1

3002

11 t t x x

200 400



PHASE DIAGRAM

xt-1

xt

xt = xt-1

3002 1 t t x x

300 400



PHASE DIAGRAM

xt-1

xt

xt = xt-1

9002 1 t t x x

300 400



EXERCISE

Consider a simple dynamic Keynesian model:

Let = ½, = 200, = 4/5, G = 300, Y0 = 2500.

(1) What is the steady state level of income ?

(2) Draw a phase diagram and show the time path of income if government spending increases to 400.

GC Y t t

1)1( t t

P

t Y Y Y

P t t Y C



SECOND-ORDER DIFFERENCE EQUATION

Consider a second-order linear difference equation

Solution to the homogeneous difference equation: Distinct roots:

Repeated roots:

Complex roots:

ybxax x t t t 21

t t

t C C x 2211

t

t t C C x )( 21

b

at

bC b

at

bC x

t t

t 2sin2cos 21



EXERCISE

Consider a simple dynamic Keynesian model:

Let = ½, = 200, = 4/5, G = 400, Y0 = 2000,

Y1 = 2100, = 1/10.

(1) What is the steady state level of income ?

(2) Derive the solution and determine the dynamics of

income.

G I C Y t t t

1)1( t t

P

t Y Y Y P

t t Y C )( 21 t t t Y Y I



EXERCISE

Consider the ff. system of difference equation:

Derive the solution.

3002.2.

9001.3.

11

11

t t t

t t t

y p y

y p p

11 difference equations

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