1 economic faculty differential equations and economic applications lesson 1 prof. beatrice venturi
TRANSCRIPT
1
Economic Faculty
Differential Equations and Economic Applications
LESSON 1prof. Beatrice Venturi
Beatrice Venturi 2
DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS
ECONOMIC ECONOMIC APPLICATIONSAPPLICATIONS
FIRST ORDER DIFFERENTIAL EQUATIONS
DEFINITION: Let • y(x) =“ unknown function”• x = free variable • y' = first derivative
Beatrice Venturi
3
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) .
Beatrice Venturi 4
FIRST ORDERDIFFERENTIAL EQUATIONS
• More generally we may consider the following equation:
• Where f is the known function.
Beatrice Venturi
5
))(,( 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.
Beatrice Venturi
),())(,( bainxallforxgxfdx
dg
1.EXAMPLE
Beatrice Venturi 7
)(xfdx
dy
)(tIdt
dK
ydx
dy
The Domar’s Growth Model
Beatrice Venturi
8
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 .
Beatrice Venturi 9
dt
tdK )(
Connection between Capital Stock and
Investment
Beatrice Venturi 10
)(tK
)(tI
Capital stock=
Investment =
)()(
tIdt
tdK
Connection between Capital and Investment
Beatrice Venturi 11
dttItdK
dttIdtdt
tdK
)()(
)()(
dttItK )()(
Connection between Capital and Investment
B eatrice Venturi 12
ctdttdttItK 2
3
2
1
23)()(cKt )0(0
)0(2)( 2
3
KttK
Connection between Capital and Investment
Beatrice Venturi 13
)()()()( aKbKtKdttI ba
b
a
1000)( tI
10001000)(1
0
1
0
dtdttI
Connection between Capital and Investment
Beatrice Venturi 14
Price adjustment in the market
• We consider the demand function:
Beatrice Venturi 15
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:
Beatrice Venturi 16
pp
)(
)(
p
Price adjustment in the market
Beatrice Venturi
17
)]()([ 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
Beatrice Venturi 18
0)( padt
dp
dtap
dp)(
ctap )(ln
Price adjustment in the market
• We use the method of integranting factors.
• We multiply by the factor
Beatrice Venturi 19
taCe )(
)(
)()(
tp
Price adjustment in the market
Beatrice Venturi 20
Solution =
)(
,))0(()(
akdove
pepptp kt
To find c put t=0
The equilibrium price P is asymptotically stable equilibrium
Beatrice Venturi 21
SEPARATION OF VARIABLES.
This differential equation can be solved by separation of variables.
ygxfy
Beatrice Venturi 22
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
)()(
)()(
)()(
)()('
Beatrice Venturi 23
The Domar Model
s(t)= marginal propensity to save is a function of t
Beatrice Venturi 24
)(1
)(
1ts
Idt
dII
tsdt
dI
0)( Itsdt
dI
dttsCetI )()(
PARTICULAR SOLUTION• DEFINITION
• The particular integral or • solution of E.D.O.
Beatrice Venturi 25
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
Beatrice Venturi 26
;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
Beatrice Venturi 27
dxxdy 2
Beatrice Venturi 28
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
Beatrice Venturi 29
52.50-2.5-5
40
20
0
-20
-40
x
y
x
y
Beatrice Venturi 30
INTEGRALE SINGOLARE
yxfy ,
We have solution that cannot be obtained by assigning a value to a the constant c.
Beatrice Venturi 31
Example:
dxdyy
dxy
dy
ydx
dy
yy
2
1
2
2
2
Beatrice Venturi 32
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.