Chapter9 Advanced Mathematics

Chapter 9 Dynamics Chapter 9 Dynamics Nguyen Thi Minh Tam ntmtam vnuagmail com December 22, 2020 1 9 1 Difference equations 2 9 2 Differential equations 9 1 Difference equations Difference equation A difference equation (sometimes called a recurrence relation) is an equation that relates consecutive terms of a sequence of numbers Example 1 Given the difference equation Yt = 3Yt−1 If Y0 = 2, what are Y1,Y2,Y3? Consider the difference equation Yt = bYt−1 + c, (1) where b,c are constants, c 6= 0 T.

Chapter 9: Dynamics

Nguyen Thi Minh Tam

ntmtam.vnua@gmail.com

December 22, 2020

1 9.1 Difference equations

2 9.2 Differential equations

9.1 Difference equations

Difference equation

equation that relates consecutive terms of a sequence of numbers

Example 1 Given the difference equation

Yt= 3Yt−1

If Y0 = 2, what are Y1, Y2, Y3?

Consider the difference equation

Yt = bYt−1+ c, (1) where b, c are constants, c 6= 0

is thecomplementary function, PS is theparticular solution The complementary function is the name that we give to the solution of equation (1) when c = 0 In this case,

CF = Abt The particular solution is the name that we give to any

solution of equation (1) In this case, we try Yt = D, D is a constant

Example 2 Solve the following difference equations with the specified initial conditions:

a) Yt = 1

4Yt−1+ 6; Y0= 1

b) Yt = −2Yt−1+ 9; Y0= 4

Note

If −1 < b < 1, then Yt converges

If b ≤ −1 or b ≥ 1, then Yt diverges

The solution of the difference equation eventually settles down

to an equilibrium state only when −1 < b < 1

If convergence does occur in an economic model, the model is said to bestable If not, it is said to beunstable

National income determination

Consumption Ct in period t depends on national income Yt−1

in the previous period t − 1 The corresponding consumption function is given by

Ct= aYt−1+ b (0 < a < 1)

If we assume that investment is the same in all time periods, then

It = I∗

If the flow of money is in balance in each time period, we have

Yt = Ct+ It

Substituting the expressions for Ct and It into this gives

Yt = aYt−1+ b + I∗

Example 3 Consider the two-sector model:

Yt = Ct+ It

Ct = 0.9Yt−1+ 250

It = 350 Find an expression for Yt when Y0 = 6500 Is this system stable or unstable?

Supply and demand analysis

The supply QS t in period t depends on the price Pt−1in the preceding period t − 1 The corresponding time-dependent supply and demand equations are

QSt = aPt−1− b

QD t = −cPt+ d where a, b, c, d are positive constants

If we assume that, within each time period, demand and supply are equal, then

QDt = QSt that is,

−cPt+ d = aPt−1− b ⇔ − cPt = aPt−1− b − d

⇔ Pt = −a

cPt−1+

b + d c

Once a formula for Pt is obtained, we can use the demand equation

Qt = −cPt+ d

to deduce a corresponding formula for Qt

Example 4 Consider the supply and demand equations

QS t = Pt−1− 8

QDt = −2Pt+ 22 Assuming that the market is in equilibrium, find expressions for Pt and Qt when P0= 11 Is the system stable or unstable?

9.2 Differential equations

Differential equation

of an unknown function

Example 5

dy

dt = 4y is a differential equation.

Consider the differential equation

dy

where m, c are constants

The general solution of equation is y = CF + PS , where

The complementary function (CF) is the solution of the equation

dy

dt = my and is given by

CF = Aemt The particular solution is any solution that we are able to find

of (2) In this case, we try y = D, D is a constant

Example 6 Solve the differential equation

dy

dt = −3y + 180 with the initial condition y (0) = 40 Comment on the qualitative behaviour of the solution as t increases

Note

If m < 0 then y (t) converges

If m > 0 then y (t) diverges

National income determination

Suppose that the rate of change of Y is proportional to the excess expenditure C + I − Y , that is,

dY

dt = α(C + I − Y ) (3) for some positive adjustment coefficientα

If C = aY + b and I = I∗, substitute into (3) we get

dY

dt = α(aY + b + I

∗− Y )

⇔dY

dt = α(a − 1)Y + α(b + I

∗)

Example 7 Consider the two-sector model

dY

dt = 0.1(C + I − Y )

C = 0.9Y + 100

I = 300 Find an expression for Y (t) when Y (0) = 2000 Is this system stable or unstable?

Supply and demand analysis

Suppose that the rate of change of price is proportional to excess demand QD− QS, that is,

dP

dt = α(QD− QS) for some positive adjustment coefficient α

If QS = aP − b, QD = −cP + d (a, b, c, d are positive constants), we have

dP

dt = α[(−cP + d ) − (aP − b)]

⇔dP

dt = −α(a + c)P + α(b + d )

Example 8 Consider the market model

QS = 2P − 2

QD = −P + 4 dP

dt =

1

3(QD− QS) Find expressions for P(t), QS(t) and QD(t) when P(0) = 1 Is this system stable or unstable?

655-656)