economic faculty

1

Economic Faculty

Differential Equations and Economic Applications

LESSON 1prof. Beatrice Venturi

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DIFFERENTIAL EQUATIONS

ECONOMIC APPLICATIONS

FIRST ORDER DIFFERENTIAL EQUATIONS

DEFINITION: Let • y(x) =“ unknown function”• x = free variable • y' = first derivative

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0),(, yxyxF

First order Ordinary Differential Equation .

FIRST ORDER DIFFERENTIAL EQUATIONS

DEFINITION: An ordinary differential equation (or ODE) is

an equation involving derivates of: y(x) (the unknown function)

a real value function (of only one independent variable x) defined in y: (a,b) Ran open interval (a,b) .

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FIRST ORDERDIFFERENTIAL EQUATIONS

• More generally we may consider the following equation:

• Where f is the known function.

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))(,( xyxfdx

dy (*)

Solution of E.D.O.

• Definition: A solution or integral curve of an EDO is a function g(x) such that when it is substituted into (*) it reduces (*) to an identity in a certain open interval (a,b) in R.

• We find a solution of an EDO by integration.

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),())(,( bainxallforxgxfdx

dg

1.EXAMPLE

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)(xfdx

dy

)(tIdt

dK

ydx

dy

The Domar’s Growth Model

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sIdt

dII

sdt

dI 11

Investment I and Capital Stock K

• Capital accumulation = process for which new shares of capital stock K are added to a previous stock .

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dt

tdK )(

Connection between Capital Stock and

Investment

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)(tK

)(tI

Capital stock=

Investment =

)()(

tIdt

tdK

Connection between Capital and Investment

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dttItdK

dttIdtdt

tdK

)()(

)()(

dttItK )()(

Connection between Capital and Investment

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ctdttdttItK 2

3

2

1

23)()(cKt )0(0

)0(2)( 2

3

KttK

Connection between Capital and Investment

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)()()()( aKbKtKdttI ba

b

a

1000)( tI

10001000)(1

0

1

0

dtdttI

Connection between Capital and Investment

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Price adjustment in the market

• We consider the demand function:

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pQd

and the supply function :

pQs

for a commodity

Price adjustment in the market

• At the equilibrium when supply balances demand , the equilibrium prices satisfies:

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pp

)(

)(

p

Price adjustment in the market

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)]()([ padt

dp

)()( apadt

dp

( )d s

dpa Q Q

dt

Suppose the market not in equilibrium initially. We study the way in which price varies over time in response to the inequalities between supply and demand.

Price adjustment in the market

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0)( padt

dp

dtap

dp)(

ctap )(ln

Price adjustment in the market

• We use the method of integranting factors.

• We multiply by the factor

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taCe )(

)(

)()(

tp

Price adjustment in the market

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Solution =

)(

,))0(()(

akdove

pepptp kt

To find c put t=0

The equilibrium price P is asymptotically stable equilibrium

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SEPARATION OF VARIABLES.

This differential equation can be solved by separation of variables.

ygxfy

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The method “ separates” the two variables y and x placing them in diffent sides of the equation:

Each sides is then integrated:

cdxxfyg

dy

dxxfyg

dy

ygxfdx

dy

ygxfy

)()(

)()(

)()(

)()('

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The Domar Model

s(t)= marginal propensity to save is a function of t

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)(1

)(

1ts

Idt

dII

tsdt

dI

0)( Itsdt

dI

dttsCetI )()(

PARTICULAR SOLUTION• DEFINITION

• The particular integral or • solution of E.D.O.

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0,, yyxF

xfy is a function :

xy obtained by assigning particular values to the arbitrary constant

Example

– Given the initial condition – the solution is unique

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;3

1;4

P

02 xy

dxxdy

xdx

dy

xy

xy

2

2

2

2

'

0'

213

213

63

3

641

3

64

3

13

4

3

1

3

3

3

3

xy

c

c

cx

y

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dxxdy 2

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52.50-2.5-5

20

0

-20

-40

-60

x

y

x

y

213

3

x

y

The graph of the particular solution

Case: C₁= 0 y=(1/3)x³

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52.50-2.5-5

40

20

0

-20

-40

x

y

x

y

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INTEGRALE SINGOLARE

yxfy ,

We have solution that cannot be obtained by assigning a value to a the constant c.

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Example:

dxdyy

dxy

dy

ydx

dy

yy

2

1

2

2

2

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2

2

1

2

1

2

1

cxy

cxy

cxy

cxy

y=0 is a solution but this solution cannot be abtained by assing a

value to c from the generale solution.