1 economic faculty differential equations and economic applications lesson 1 prof. beatrice venturi
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1
Economic Faculty
Differential Equations and Economic Applications
LESSON 1prof. Beatrice Venturi
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DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS
ECONOMIC ECONOMIC APPLICATIONSAPPLICATIONS
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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 .
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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 (*)
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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
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1.EXAMPLE
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)(xfdx
dy
)(tIdt
dK
ydx
dy
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The Domar’s Growth Model
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sIdt
dII
sdt
dI 11
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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 )(
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Connection between Capital Stock and
Investment
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)(tK
)(tI
Capital stock=
Investment =
)()(
tIdt
tdK
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Connection between Capital and Investment
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dttItdK
dttIdtdt
tdK
)()(
)()(
dttItK )()(
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Connection between Capital and Investment
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ctdttdttItK 2
3
2
1
23)()(cKt )0(0
)0(2)( 2
3
KttK
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Connection between Capital and Investment
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)()()()( aKbKtKdttI ba
b
a
1000)( tI
10001000)(1
0
1
0
dtdttI
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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
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Price adjustment in the market
• At the equilibrium when supply balances demand , the equilibrium prices satisfies:
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pp
)(
)(
p
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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.
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Price adjustment in the market
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0)( padt
dp
dtap
dp)(
ctap )(ln
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Price adjustment in the market
• We use the method of integranting factors.
• We multiply by the factor
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taCe )(
)(
)()(
tp
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Price adjustment in the market
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Solution =
)(
,))0(()(
akdove
pepptp kt
To find c put t=0
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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:
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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 )()(
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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
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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'
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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
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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.