Mathematical Methods in Economics · We shall begin with continuous-time systems of di⁄erential...

Mathematical Methods in EconomicsLecture #3

[email protected] () www.wne.uw.edu.pl/rkruszewski/mmine 1 / 25

Required readings !Ronald Shone �Economic Dynamics" :Lecture #3 ! 2.1, 2.2, 4.1, 4.4, 4.5, 4.6, 4.7, 4.8


We shall begin with continuous-time systems of di¤erential equations,which typically take the canonical form

dxdt= x 0 =

�x = f (x) (1)

where f is a function with domain U an open subset of Rm and range Rm

f : U ! Rm

The vector

x = (x1, x2, ..., xm)T

denotes the physical variables to be studied, or some appropriatetransformations of them; t 2 R indicates time. The variables xi aresometimes called dependent variables whereas t is called theindependent variable.Equation (1) is called autonomous when the function f does not dependon time directly, but only through the state variable x .


The space Rm , or an appropriate subspace of dependent variables � thatis, variables whose values specify the state of the system � is referred toas the state space. It is also known as the phase space or, sometimes,the con�guration space. In simple, low-dimensional graphicalrepresentations of the state space the direction of motion through time isusually indicated by arrows pointing to the future. The enlarged space inwhich the time variable is explicitly considered is called the space ofmotions.The function f de�ning the di¤erential equation (1) is also called a vector�eld, because it assigns to each point x 2 U a velocity vector f (x). Asolution of (1) is a di¤erentiable function x(t) which substituted into theequation satis�es it exactly in some interval.

x : I ! Rm

and I is an interval of R. A general solution to a di¤erential equation is asolution, whether expressed explicitly or implicitly, which contains allpossible solutions over an open interval. If we want to indicate that thesolution passes through the initial point x0 at time t0 we write x(t0) = x0.


Theorem

Let�x = f (t, x) be continuous with continuous derivative ∂f

∂x for x indomain D and t in interval I, i.e. for x 2 D and t 2 I . Then if x0 2 D andt0 2 I , there exists a solution x�(t) de�ned uniquely in someneighbourhood of (x0, t0) satisfying the initial condition x�(t0) = x0.

The set of solution curves of a dynamical system, sketched in the statespace, is known as the phase diagram or phase portrait of the system.Phase diagrams in two dimensions usually provide all the information weneed on the orbit structure of the continuous-time system.


ExampleLet us consider the equation

�x = x

Picture 1a shows a vector �eld of this equation and the picture 1b showsgraphs of the solution for eight di¤erent initial conditions

Picture 1a Picture [email protected] () www.wne.uw.edu.pl/rkruszewski/mmine 6 / 25

De�nitionEquation

x (n) = f (x ,�x ,��x , ..., x (n�1))

is an autonomous, ordinary di¤erential equation of order n, where n isthe highest order of di¤erentiation with respect to time appearing in theequation. It can be always put into the canonical form (1) by introducingan appropriate number of auxiliary variables.


De�nitionThe typical �rst order linear di¤erential equation is of the form

�x + p(t)x = q(t) (2)

where p(t) and q(t) are continuous function on an open interval (a, b).

De�nitionIf q(t) � 0 then equation (2) is called the homogeneous linear di¤erentialequation.

De�nitionIf q(t) 6� 0 then equation (2) is called the non homogeneous lineardi¤erential equation.


The four-step solution procedure of (2) is as follows:Step 1 Write the linear �rst-order equation in the standard form

�y + p(t)y = q(t)

Step 2 Calculate the integrating factor

µ(t) = eP (t)Zp(t)dt = P(t) + C

Step 3 Multiply throughout by the integrating factor, integrateboth sides

µ(t)�dydt+ p(t)y

�= µ(t)qt)

or

ddt[µ(t)y ] = µ(t)q(t)

Z ddt[µ(t)y ] dt =

Zµ(t)q(t)dt


Step 4 Write the general result

µ(t)y =Z

µ(t)q(t)dt

or

y(t) =1

µ(t)(G (t) + C )

where G 0(t) = µ(t)q(t)

ory(t) = e�P (t)

Zq(t)eP (t)dt

Example

Find the general solution of�x + 2x

t = 5t2.


De�nitionThe equation

�x + p(t)x = q(t)xn (n 6= 0, 1) (3)

is called the Bernoulli equation, after James Bernoulli who solved it in1695.

It is non linear, because of the presence of xn, but could be made linear bydividing through by xn and de�ning the new variable y = x1�n

x�n�x + p(t)x1�n = q(t)

or, in the new variable y ,

11� n

�y + p(t)y = q(t)

�y + (1� n)p(t)y = (1� n)q(t)

Example

Find the general solution of�x � x

t = �t2x3.


Let us consider linear dynamical system on the plane

�x = Ax =

�a11 a12a21 a22

� �x1x2

�(4)

with x1, x2 2 R, aij real constants.First, notice that the functions x1 = 0, x2 = 0 solves (4) trivially. Thisspecial solution is called the equilibrium solution, because if x1 = 0 andx2 = 0 then

�x1 = 0 and

�x2 = 0

That is, a system starting at equilibrium stays there forever. Notice that ifA is nonsingular, x1 = 0 and x2 = 0 is the only equilibrium for linearsystems like (4). The di¤erent types of dynamic behaviour of (4) can bedescribed in terms of the two eigenvalues of the matrix A, which inplanar systems can be completely characterized by the trace anddeterminant of A.


Characteristic equation for matrix A is det (A� λI ) = 0

det��

a11 a12a21 a22

�� λ

�1 00 1

��= det

��a11 � λ a12a21 a22 � λ

��=

= λ2 � (a11 + a22)λ+ (a11a22 � a12a21) = λ2 � tr(A)λ+ det(A) = 0and the eigenvalues are

λ1,2 =12(tr(A)�

p∆)

where

∆ = [tr(A)]2 � 4 det(A)is called the discriminant.In the following we consider nondegenerate equilibria (for which λ1 andλ2 are both nonzero). We distinguish behaviours according to the sign ofthe discriminant.


Case 1 ∆ > 0. Two real distinct eigenvalues λ1,λ2 with real eigenvectorsv1, v2. The solution of (4) is

x(t) = c1v1eλ1t + c2v2eλ2t

where c1 and c2 are real constants.We have three basic sub cases:


(i) tr(A) < 0, det(A) > 0 in this case, eigenvalues are negative. Theequilibrium is called a stable node.

(λ1 < λ2 < 0) (λ2 < λ1 < 0)


(ii) tr(A) > 0, det(A) > 0. In this case, eigenvalues are positive. Theequilibrium is called an unstable node.

(0 < λ2 < λ1) (0 < λ1 < λ2)


(iii) det(A) < 0. In this case, which implies ∆ > 0 independently of thesign of the trace of A, one eigenvalue is positive, the other is negative.The equilibrium is known as saddle point.

(λ1 < 0 < λ2) (λ2 < 0 < λ1)

Saddle point is unstable, but as we can see on the phase portraits thereexists one direction which converges to the equilibrium (so called stableseparatrice).


Mathematically speaking, a saddle point is unstable. In economicliterature, especially in rational expectations or optimal growth models, weoften �nd the concept of saddle point stability which, in the light of whatwe have just said, sounds like an oxymoron. The apparent contradiction isexplained by considering that those models typically comprise a dynamicalsystem characterized by a saddle point, plus some additional constraintson the dynamics of the system such that, given the initial condition ofsome of the variables, the others must be chosen so as to position thesystem on the stable separatrice. Therefore, the ensuing dynamics isconvergence to equilibrium.


Case 2 ∆ < 0 The eigenvalues and eigenvectors are complex conjugatepairs. λ1 = α+ iβ, λ2 �

_λ1 = α� iβ with corresponding eigenvectors v1

and v2(=_v 1). The solution of (4) is

x(t) = c1v1eλ1t + c2v2eλ2t

= c1v1e(α+iβ)t + c2v2e(α�iβ)t

= (c1v1 + c2v2)eαt cos βt + i(c1v1 � c2v2)eαt sin βt= eαt (h1 cos βt + h2 sin βt)

where h1 = c1v1 + c2v2 and h2 = i(c1v1 � c2v2) both real vectors.


(i) tr(A) < 0, (Re(λ1,2) < 0). The oscillations are dampened and thesystem converges to equilibrium. The equilibrium point is known as afocus (stable focus) or, sometimes, a vortex (stable vortex), due to thecharacteristic shape of the orbits around the equilibrium.


(ii) tr(A) > 0, (Re(λ1,2) > 0). The amplitude of oscillations gets largerwith time and the system diverges from equilibrium. The equilibrium pointis called an unstable focus.


(iii) tr(A) = 0, (Re(λ1,2) = 0), det(A) > 0. In this special case we have apair of purely imaginary eigenvalues. Orbits neither converge to, nordiverge from, the equilibrium point, but they oscillate regularly around itwith a constant amplitude that depends only on initial conditions and afrequency equal to p

det(A)2π

The equilibrium point is called a centre. We shall have more to say on thisspecial case when we discuss nonlinear systems.


Case 3. ∆ = 0. The eigenvalues are real and equal λ1 = λ2 = λ. If matrixA has two independent eigenvectors v1 and v2, then the solution of (4) is

x(t) = c1v1eλt + c2v2eλt

where c1 and c2 are real constants.If matrix A has only one eigenvector v1 then the solution is

x(t) = c1h1(t) + c2h2(t)

whereh1(t) = eλtv1 where v1 satis�es (A� λI )v1 = 0h2(t) = eλt (tv1 + v2) where v2 satis�es (A� λI )v2 = v1


In this case, if A 6= λI , only one eigenvector can be determined. Theequilibrium point is again a node, sometimes called a Jordan node (stablefor λ < 0 and unstable for λ > 0). An example (for λ > 0) of this type isshown on picture (a). Finally, if A = λI the equilibrium is still a node,sometimes called a bicritical node (or a star). Example of this type forλ > 0 is shown on picture (b).

(a) (b)[email protected] () www.wne.uw.edu.pl/rkruszewski/mmine 24 / 25

Finally the following picture provides a very useful geometricrepresentation in the

(tr(a), det(a))

plane of the various cases discussed above.


Mathematical Methods in Economics · We shall begin with continuous-time systems of di⁄erential...

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