LECTURE NOTES ON ADVANCED …mathematical-methods.wikispaces.com/file/view/MM.pdf2 ADVANCED MACROECONOMICS I - WS 2011/2012 1. ... ñ solution is tied down to the present ñ solution

LECTURE NOTES ON ADVANCED MACROECONOMICS I

(MATHEMATICAL METHODS)

Date: November 01, 2011.

1

2 ADVANCED MACROECONOMICS I - WS 2011/2012

1. The most elementary dynamics of love and hate

1.1. The daily love dynamics of Romeo and Juliet. Partial love mechanism(updating procedure) where Rt “ measures the degree of affection

Rt`1 “ Rt ` τ0: ÝÑ (additional) appeal p`q/repultion p´q that Juliet pJqpresents to Romeo pRq, in absence of other feelings

Rt`1 “ Rt ` bJt: ÝÑ extent to which Romeo pRq is en-/discouraged byJuliet’s pJ 1sq love

Rt`1 “ Rt ` aRt: ÝÑ extent to which Romeo pRq is en-/discouraged by hisown feelings (love to love)

Hence, there are 4 possible romantic styles of R and J such like

a, b ą 0 (“eager beaver”)

a ą 0 ą b (“narcistic nerd”)

a ă 0 ă b (“cautious lover”)

a, b ă 0 (“hermit”)

These relationships can be aggregated to the following dynamic equations whichrepresent a deterministic dynamic system (linear 2D difference equation system)

Rt`1 “ Rt ` aRt ` bJt (1)

Jt`1 “ Jt ` cRt ` dJt (2)

D a point of rest (D an equilibrium) pR˚, J˚q? ÝÑ where is Rt`1 “ Rt true formultiple periods?Answer: Yes, there is an equilibrium at pR˚, J˚q “ p0, 0q, but are there otherEquilibriums (additional non-zero solutions)?

p1q aR˚ ` bJ˚ “ 0 ñ R˚ “ ´b

aJ˚

p2q cR˚ ` bJ˚ “ 0 ñ R˚ “ ´d

cJ˚

ùñ Amount of equilibria “#

ba

‰ dc

unique equilibriumba

“ dc

continuum of equilibria

1.2. High frequency love dynamics. So far: Reaction to each other only oncea day (daily love dynamics with h “ 1).Now: Reaction to each other more than once a day (updating more frequently, everyh days with h ď 1, e.g. every hour with h “ 1

24 ) describes a dynamic system indiscrete time (DT).

Rt`h “ Rt ` h raRt ` bJtslooooomooooon

adjustment term

(3)

Jt`h “ Jt ` hrcRt ` dJts with h as adjustment period (4)

Now we can rewrite the system to

Rt`h ´ Rt

h“ aRt ` bJt

Jt`h ´ Jt

h“ cRt ` dJt

Additionally we change the notation like Rt ÝÑ Rptq for the rest of the lecturenotes. To get from a dynamic system in discrete time to a dynamic system incontinuous time (CT) we have to go to the limit

limhÑ0

Rpt ` hq ´ Rptqh

“ dRptqdt

“ R1ptq “ 9Rptq

ADVANCED MACROECONOMICS I - WS 2011/2012 3

Thus, the derivative of the dynamic system in discrete time (with respect to time)gives us the dynamic system in continuous time (2D differential equation system)

9Rptq “ aRptq ` bJptq omit timeÝÝÝÝÝÝÑ 9R “ aR ` bJ (5)

9Jptq “ cRptq ` dJptq omit timeÝÝÝÝÝÝÑ 9J “ cR ` dJ (6)

Economy in DThigh frequency adjustmentsÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÑ h-economy

limhÑ0ÝÝÝÝÑ economy in CT

Solution Concept:

DT) find a sequence tpRt, Jtqu8t“0 satifying (1) and (2)

CT) find a differentiable functions R, J “ r0,8q Ñ R satisfying (5) and (6)@ t ą 0How to obtain a solution in CT:1. solution path of (5) and (6) can be analytically calculated (math.

theory, rarely done); qualitative insights suffice

2. more complex (non-linear)oftenÝÝÝÑ no explicit solution available; quali-

tative insights perhaps possible3. simulation - go backwards from (5) and (6): simulate h-economy (just

iterate) with h “small enough” to get an approximation

Which method is preferable (or more appropriate to the real world)?

‚ high frequency decisions in real world? (infrequent decisions, but manyagents)

‚ often CT better analytical traceable‚ mostly: not matter of principle but of convenience

1.3. Explorations of different scenarios.

phase plane monotonic cyclicalconvergence (stable) SM SCdivergence (unstable) UM UC

Using a parameter plane (see 2): Fix all parameters of the system except two, findout which combinations of the latter give risc to what type of behavior.

1.4. Adaptive vs. rational expectations love dynamics.

dated state of the artfeedback guided: intertemporal optimizationR feeds back on J and vice-versa rational expec. (RE) or rational choice (RC)adaptive behavior:blind, ignorant to progress in economic theory

Rational Choice.

‚ At day t, R solves his happiness-problem pT “ 8q- given his (rational) expectations: EtRt`τEtJt`τ with τ “ 0, 1, 2 . . .8- yields optimal affection Rt

‚ At day t, J solves her happiness-problem pT “ 8q- given her (rational) expectations: EtRt`τEtJt`τ with τ “ 0, 1, 2 . . .8- yields optimal affection Jt

ùñ Let’s suppose the optimality conditions can be reduced to a certain type ofrelationships (Euler conditions):

Rt “ aEtRt`1 ` bEtJt`1 ` εR,t

Jt “ cEtRt`1 ` dEtJt`1 ` εJ,t


R

J

cyclical convergence

monotonic convergence

Figure 1. Convergence

Parameter 2

Parameter 1

UC

SC

SM

UM

unstable

stable

cyclical

monotonic

Figure 2. Parameter plane

Here the absence of random shocks is assumed so that RE implies perfect foresightwhich leads to EtRt`1 “ Rt`1.

Rt “ aRt`1 ` bJt`1

Jt “ cRt`1 ` dJt`1 (7)


Note that in contrast to the backward-looking system (1) and (2) the system is nowforward-looking. We can convert the system into matrices

xt “„

Rt

Jt

, A “„

a b

c d

so that

xt “ Axt`1

A´1xt “ xt`1

with A´1 “ 1ad´bc

„

d ´b

ć a

. Hence, (7) yields

Rt`1 “ αRt ` βJt

Jt`1 “ γRt ` δJt (8)

Is the system (8) equal to the system (1) and (2)? ùñ No!

(1) and (2) (8)

Rt`1 “ Rt ` aRt ` bJt Rt`1 “ αRt ` βJt

Jt`1 “ Jt ` cRt ` dJt Jt`1 “ γRt ` δJt

Rule of game of (1) and (2): Rule of game of (8):- initial conditions R0, J0 are historically given - initial conditions R0, J0 are part of the solution(then simply iterate forward)- (Rt, Jt) are predetermined - no predetermined variables

- additional requirement, amounts to theterminal condition R8 “ 0, J8 “ 0

ñ solution is tied down to the present ñ solution is tied down to the infinite future

‚ Backward-looking (with intrinsic dynamics):= system is determined by thepast (by things that already happend or known values) Ñ need of an initinalcondition

‚ Forward-looking (no intrinsic dynamics):= system determined by the future(by things that will happen or by yet unknown values) Ñ there’s no initialcondition; RE means forward-looking system

– no shocks: (under “determinacy”) at t “ 0 agents jump into R0 “0, J0 “ 0

– with shocks: movements are entirely caused by shocks (shock-driven)

2. (Linear) Difference equation systems

2.1. Types of DT systems and some relashionships among them.

(1) Inhomogenous linear n-th order (scalar) difference equation

yt ` a1yt´1 ` a2yt´2 ` . . . ` anytń ` y “ 0

with y, y P R [e.g. VAR= vector autoregressive system](1a) Homogenous linear n-th order (scalar) difference equation (if y “ 0)

yt ` a1yt´1 ` a2yt´2 ` . . . ` anytń “ 0

(2) Homogenous (autonomous) linear n ˆ n difference equation system

xt`1 “ Axt or xt “ Axt´1

with x P Rn, A P Rnˆn

(3) Inhomogenous (autonomous) linear n ˆ n difference equation system

xt`1 “ Axt ` x or xt “ Axt´1 ` x

with x, x P Rn, A P Rnˆn


(4) General (autonomous) non-linear difference equation system

xt`1 “ F pxtqwith F “ Rn Ñ Rn

(5) General (non-autonomous) non-linear difference equation system

xt`1 “ F pt, xtqwith F “ R ˆ R

n Ñ Rn

Existence and uniqueness of solution is no problem once an initial condition is given.

Equilibrium notion of (1)“Equilibrium”, “Point of rest”, “stationary point” are used interchangeably for apoint where the system reproduces itself over time.

y˚ is an equilibrium of (1) :ðñ yt “ yt´1 “ . . . “ ytń “ y˚

y˚ p1 ` a1 ` . . . ` anqlooooooooooomooooooooooon

‰0

`y “ 0

y˚ “ ý

1 ` a1 ` . . . ` an

Equilibrium notion of (3)

x˚ is an equilibrium of (3) :ðñ xt`1 “ xt “ x˚

x˚ “ Ax˚ ` x

pI ´ Aqx˚ “ x

x˚ “ pI ´ Aq´1x

pI ´ Aq is invertible by hypothesis.

Proposition 2.1. If its equilibrium is unique, system (1) can be transformed intoa homogenous n-th order difference equation (1a).

Proof.

zt “ yt ´ y˚

Claim: tytu8t“0 satisfies (1) ðñ tztu8

t“0 satisfies (1a)write yt “ zt ` y˚ in (1) and factorize

zt ` y˚ ` a1pzt´1 ` y˚q ` . . . ` anpztń ` y˚q ` y “ 0

zt ` a1zt´1 ` . . . ` anztń ` y˚p1 ` a1 ` . . . ` anq ` ylooooooooooooooomooooooooooooooon

“0

“ 0 (6)

“ (1a) q.e.d.

Proposition 2.2. A homogenous n-th order difference equation (1a) can be trans-formed into a n ˆ n difference equation system (2).


Proof. Let (1a) ðñ (6) be given. Def

x1,t “ ztń

x2,t “ x1,t`1 “ ztń`1

...

xn,t “ xn´1,t`1 “ zt´1

with xn,t “ n´th component of the vector x at period t.(1a) ðñ (6) ðñ zt “ á1zt´1 ´ . . . ´ anztń:

xn,t`1 “ zt “ á1xn,t ´ a2xn´1,t ´ . . . ´ anx1,t

xn´1,t`1 “ xn,t

xn´2,t`1 “ xn´1,t

...

x2,t`1 “ x3,t

x1,t`1 “ x2,t

xt`1 “ Axt

with

A “

»

—

—

—

—

—

–

0 1 0 . . . 00 0 1 . . . 0...

......

. . ....

0 0 0 . . . 1án án´1 án´2 . . . á1

fi

ffi

ffi

ffi

ffi

ffi

fl

q.e.d.

Proposition 2.3. Generally also a pnˆnq´difference equation system (2) can betransformed into a homogenous n´th order difference equation (1).

Proof. Consider n=2

x1,t`1 “ a11x1,t ` a12x2,t (a)

x2,t`1 “ a21x1,t ` a22x2,t (b)

‚ If a12 “ a21 “ 0 ñ two isolated equations ñ each equation can be treatedon its own.

‚ If a12 or a21 are non-zero, paq and pbq can be transformed into p1aq.Assume: a12 ‰ 0

(a) ðñ x2,t “ 1

a12x1,t`1 ´ a11

a12x1,t (c)

going one period forward

ðñ x2,t`1 “ 1

a12x1,t`2 ´ a11

a12x1,t`1 (d)

insert (d) in LHS of (b), and (c) in RHS of (b)

1

a12x1,t`2 ´ a11

a12x1,t`1 “ a21x1,t ` a22

a12x1,t`1 ´ a22a11

a12x1,t

ˇ

ˇ ¨ a12x1,t`2 ´ pa11 ` a22qx1,t`1 ´ pa12a21 ´ a22a11qx1,t “ 0

q.e.d.


Proposition 2.4. If (3) has a unique equilibrium, (3) can be reduced to (2) withrespect to x˚ “ pI Áq´1x ¨ fpzq and zt “ xt ´ x˚: suppose txtu solves (3) ñ thentztu solves (2).

Proof. Equilibrium point: x˚ “ pI ´ Aq´1x pñ x “ pI ´ Aqx˚qi) we have

zt`1 “ Azt ðñ pxt`1 ´ x˚q “ Apxt ´ x˚qðñ xt`1 “ Axt ` x˚ ´ Ax˚ “ Axt ` pI ´ Aqx˚ “ Axt ` x

ii)

xt`1 ´ x˚ “ Axt ` x ´ x˚ “ Axt ´ x˚ ` pI ´ Aqx˚

“ Axt ´ Ax˚ ´ x˚ ` x˚

“ Apxt ´ x˚qzt`1 “ Azt

Conclusion: Given non-homogenous autonomous equation system (1) and (3), inorder to solve it, do the following:

(1.) Calculate the equilibrium value x˚, y˚

(2.) Compute the solution of the homogenous equation system xH , yH

(3.) Calculate the general solution of (1a) or (3) by combining the above solu-tions

yG “ yH ` y˚

xG “ xH ` x˚

x2

x1

z2

z1Shift

z2

z1


2.2. Homogeneous linear difference equation systems (HDE).

Proposition 2.5 (Superposition Theorem). Let txtu, tytu be two solutions of HDEand tztu “ taxt ` bytu for some a, b P R. Then also tztu is a solution of HDE.

Proof.

zt`1 “ axt`1 ` byt`1 pwith xt`1 “ Axtq“ aAxt ` bAyt

“ Apaxt ` bytq“ Azt

q.e.d.

Remark

i) Can be extended to linear combinations of arbitrarily many solutionsii) Also holds for complex valued solutions of HDE

ùñ Select “nice” solutions and construct from them new solutions. Conversely,decompose a general solution into a linear combination of these “nice” solutions.For n “ 1 :

xt`1 “ axt for a, x P R

xt “ axt´1 “ apaxt´2q “ a2xt´2 “ a3xt´3 “ . . . “ atx0

Proposition 2.6. Given an initial condition x0 P Rn, the solution of HDE isgiven by xt “ Atx0.

Proof. Generalize case n “ 1!

Hence, eigenvalues are the core of the theory that is build in this lecture the nextpart deals with this important instrument.

Definition 1. λ P C is an eigenvalue (EV) of A and v P Cn an eigenvector ifAv “ λv

Remark

i) includes EV’sii) if λ P R ùñ v P Rn

iii) if v is an eigenvector ùñ any multiple of v is an eigenvector too

Proposition 2.7. If λ P C is an EV with eigenvector v P Cn then, starting out

from an initial vector x0 “ v leads to

xt “ λtv

in order to reduce the n-dimensional to a one-dimensional problem.


Observe

Av “ λv

A2v “ apAvq “ Aλv “ λAv “ λλv “ λ2v

A3v “ λ3v

...

Atv “ λtv

then it follows that

xtProp. 6“ Atx0 “ Atv

“ λtv

“ λtx0

Therefore, real-valued EV’s (λ P R) give rise to monotonistic motions on a straightline through the origin.

x2

x1

>1 = divergence

x2

x1

0<<1 = convergence

v1

v2

Figure 3. Phase plane

λ ą 1 ùñ divergence

λ ă 1 ùñ convergence

λ “ 1 ùñ possibly none of that; more information is needed

If λ P C it has the following form: λ “ pα, βq “ α ` iβ and λc “ pα,´βq “ α ´ iβ

(conjugate complex).

D a radius r ě 0 and an angle θ P R such that |λ| “ r “ a

α2 ` β2 (“absolutevalue” or “modulus”) and

α “ r ¨ cospθqβ “ r ¨ sinpθq

it follows that if λ P C is an EV λc is an EV too.The appropriate eigenvectors associated with such EV λ, λc are defined by

v “ w ` iz

vc “ w ´ iz

with w, z P Rn


Proposition 2.8. Let λ, λc “ α ˘ iβ “ rpcospθq ˘ i sinpθqq be a pair of conjugatecomplex EVs of A with eigenvectors v, vc “ w ˘ iz (w, z P Rn). Then for anyc1, c2 P R

xt “ rt rcosptθq rc1w ` c2zs ` sinptθq rc2w ´ c1zss (9)

is a real-valued solution of HDE.

Note: Initial condition given by x0 “ c1w ` c2z where the first part is real and thesecond one imaginary.Significance: Complex EVs give rise to cyclical motions, i.e. sin cos-expressionslead to cyclical dynamics.Thus, if

‚ |λ| “ r ă 1 we get spiralling convergence towards the zero vector‚ |λ| “ r ą 1 we get outward spiralling divergence

(Concerning the exam: It won’t be required to know the proof and the expres-sion given, i.e. the sin cos-expressions above, but of course the stylized facts.)

For business cycles in a model the period is of special interest. Complex EVswill give us information regarding this. First, we start with its definition:

Definition 2. A function of time f : Rt Ñ Rn is periodic with period T (thelength of a period) if the system at time t ` T is the same at all points in time

fpt ` T q “ fptq @t ě 0

and T is the smallest positive number with that property.

Consider:

beginning end1) General argument x Ñ sinpxq x “ 0 x “ 2π Ñ sinp0q “ sinp2πq2) With time interpretation t Ñ fptq : sinptθq t “ 0 t “ T Ñ sinp0q “ sinpTθqHence, it holds that

Tθ “ 2π

and the period is determined by

T “ 2π

θwith θ “ arcsin

«

|β|a

α2 ` β2

ff

Example (Love Dynamics): A “„´1.50 ´1.001.00 1.00

The EVs 1&2 are complex and

their radius r is equal to one (|λ| “ r “ 1). Thus, we have strictly periodicmovements on a circle (with radius r) around the origin.

Proposition 2.9 (regular case). Suppose matrix A P Rnˆn has n distinct (nomultiple) EVs. Then for any initial vector x0 P Rn there is a uniquely deter-

mined set of coefficients c1, c2, . . . , cn such that the solution txtu of HDE is thecorresponding linear combination of the expressions in proposition 7 and 8.

Proof. See [5]

If there are multiple EVs (announcement):

‚ more complicated‚ |λ| ă 1 ñ convergence

|λ| ą 1 ñ divergencebut |λ| “ 1 ñ divergence! So while for single eigenvalues with r=1 we wereon a closed orbit, the system will now diverge.


x2

x1

t=0 with f(t)=f(t+T)

t=1

t=2

t=3

t=4

t=5t=6

t=7

Figure 4. Periodic function of time with period T

Proposition 2.10. Let λ be an EV of “multiplicity” m pď nq. Then the solutiontxtu to which this EV gives rise (analogously to prop. 7 and 8) is given by

xt “ λt

»

—

—

—

–

p1ptqp2ptq...

pnptq

fi

ffi

ffi

ffi

fl

, (10)

where p1ptq, . . . , pnptq are polynomials in t of order less than or equal to m´ 1 (i.e.p1ptq “ a0 ` a1t` . . .` am´1t

m´1 which includes the expression for eigenvalues weknow from earlier on.).

Remark

i) λ may also be complex. Then pip¨q are complex-valued polynomials which,however, can be treated analogously to Prop. 8.

ii) m “ 2, λ P R Ñ D vectors v, w P Rn such that

xt “ λt

»

—

–

v1 ` w1t...

vn ` wnt

fi

ffi

fl

with vn “ ordinary eigenvectors. The w’s can sloppily be called t-generalizedeigenvectors (this is not exactly the stricter mathematical concept of gen-eralized EVs).

Proposition 2.11. Any solution of HDE can be uniquely expressed as a linearcombination of the (real-valued) characteristic solutions (7), (8), (10).


Example (Love Dynamics): A “„

1.10 ´1.001.10 ´1.00

with λ1 “ 1.370 and λ2 “ 0.730.

xt “ c1λt1 `c2λ

t2 will have a divergent part with λ1 ą 1 as well as a convergent part

with λ2 ă 1. The effect will be dominated by the first EV λ11 and more specifically

the eigenvector. Sooner or later (with increasing t) the motion will be closer to theflatter line.

If as before we have e.g. 2 real EVs. xt “ c1λt1v1 ` c2λ

t2v2 can represent any

solution (with c1, c2 as scalars which defines the linear combination). This worksbecause v1 and v2 are linearly independent.Application using the figure above: Will the agents be stuck with their starting

values and converge/diverge without control about the process? No! Think of x1

as capital and x2 as consumption. Then the problem is somewhat different fromthe maths as agents will be able to choose their consumption for the next periodand thus jump onto the stable manifold (the line indicating convergence). We willcall this saddle point (in)stability. In part 2 of advanced macro we will take a lookat the Dornbusch model which will use this feature in a continous setting.

2.3. Stability conditions for autonomous discrete-time systems. There is ahandout concerning this topic. The information from that handout has to be fullyknown. The exam will be somewhat an open-book one. One sheet of paper canbe prepared in advance by everyone with no regulations regarding font size, content(or size?).

Definition 3. Consider an n-dimensional difference equation system of the form

xt`1 ` F pxtq (11)

where F : Rn Ñ Rn is continously differentiable. Let x˚ P R

n be a satisfactory (orfixed) point of our system (11) satisfying x˚ “ F px˚q. This equilibrium is called

a) locally asymptotically stable (LAS) if every trajectory starting sufficiently

close to x˚

– stays in a close neighbourhood of x˚

– and converges to x˚ as t Ñ 8b) globally attracting if every trajectory goes towards x˚ as t Ñ 8

1Because λ1 is the numerically greater root (since 1.370 ą 0.730).


b’) globally asymptotically stable (GAS) if LAS and attracting

c) locally stable if every trajectory starting sufficiently close to x˚ stays in theclose neighbourhood

d) unstable if it is not locally stable.

Points, systems or matrices can all be stable.

Proposition 2.12. If the mapping F p¨q is linear then LAS automatically meansGAS.

Proof. Convergence for every initial point “close” to x˚ also means we have con-vergence for every initial point that is a multiple of x˚. Hence, x0 will show thesame behavior as αx0.

(It’s easy to see this when looking at the graph above.)

Definition 4. Let x˚ be an equilibrium point of (11). Its Jacobian matrix

J P Rnˆn is given by

J “

»

—

–

j11 . . . j1n...

. . ....

jn1 . . . jnn

fi

ffi

flwith jij :“ BFipx˚q

Bxj

(12)

with i, k “ 1, . . . , n

The Jacobian matrix at the equilibrium point x˚ is evaluated because the bavaviorof the system near an equilibrium point is related to the EVs of the Jacobian matrixof F at the equilibrium point. If all EVs have a negative real part the system isstable in the operating point. If any EV has a positive real part then the point isunstable (occasionally J is denoted as A in other books, so don’t be confused).

Definition 5. System (11), linearised around x˚, is given by

xt`1 “ x˚ ` Jpx ´ x˚qRemark

i) If F is linear (similar to (2), (3) in the first lectures) then BFBx “ J “ A

(This simply means that J is a matrix A, nothing deeper.).ii) (Fundamental!) If F is non-linear then, nevertheless, locally the dynamics

of the general system of (11) are “qualitatively” the same as that of thelinearised system.

Proposition 2.13. Let x˚ be an equivalent of (11) and J its Jacobian matrix.Then x˚ is LAS if the corresponding linear HDE xt`1 “ Jxt is LAS, i.e. if |λ| ă 1 @EVs of J .

Consider

λ P C is an EV of A

ðñ Dv P Cn with Av “ λv

ðñ λIv ´ Av “ 0

ðñ pλI ´ Aqv “ 0

ðñ Av ´ Iv “ 0

For this to hold, the emerging matrix must be singular (not invertible) which iseasiest expressed by

detpλI ´ Aq “ 0 ðñ#

detpλI ´ Aq “ 0

detpA ´ λIq “ 0


linearizedversion originalversion

and this is called the characteristic equation.Observation:

P pλq :“ detpλI ´ Aq

is an n-th order polynomial in λ and is called the characteristic polynomial. λis called a root of the polynomial P p¨q.RemarkA matrix A and its transpose A1 have the same characteristic polynomial and thusthe same EVs.

Proposition 2.14. Let x˚ P Rn be an equilibrium of (11) and let J be its Jacobianmatrix then

a) the equilibrium is locally asymptotically stable (LAS) if all roots λ of thecharactersitic polynomial are less than 1 in modulus/absolute value

b) x˚ is unstable if at least one root exceeds unity (in modulus)c) if F is non-linear (proper, i.e. no linearities) and if all EVs are less than

1 except one EV (or several) the modulus of which is equal to one then thelocal stability of x˚ cannot be determined from the Jacobian alone.


Proposition 2.15 (Weak conditions, see Gandolfo p. 117 or the handout).Let A P Rnˆn and for j “ 1, . . . , n define

Sj :“n

ÿ

i“1

aij (column sums)

Saj :“

nÿ

i“1

|aij | (“absolute” column sums)

Then

a) Saj ă 1 @j ñ A is asymptotically stable (AS) [sufficient cond. for stability]

b) If AS ñ ˇ

ˇ

řni“1 aii

ˇ

ˇ ă n [necessary condition for stability]

c) If AS ñ ˇ

ˇ detAˇ

ˇ ă 1 [necessary condition for stability]d) Sj ą 1 @j ñ A is unstable [sufficient condition for instability]

The logical order of the arguments (contraposition):

stable ñ condition x

not (stable) ð not (condition) x

Now let’s move to the two-dimensional case n “ 2 with

P pλq “ . . . “ λ2 ´ ptrAqλ ` detA!“ 0.

The pq´formula gives λ1,2 “ 12 trA ˘ ?

∆ where ∆ stands for the discriminant.The information whether the system is cyclical can be gotten from the informationregarding complex EVs and hence the discriminant (Homework: under what con-dition does a saddle point occur?). An unrelated though helpful hint regarding ashorter notation is to write the following:

P pλq “ λ2 ` a1λ ` a2!“ 0

with a1 “ ´ trA and a2 “ detA.Next we introduce sufficient as well as necessary conditions for stability.

Proposition 2.16 (Schur conditions for n “ 2).The following inequalities are sufficient as well as necessary for the matrix A to beasymptotically stable

a) 1 ` a1 ` a2 ą 0

b) 1 ´ a1 ` a2 ą 0

c) 1 ´ a2 ą 0

Proposition 2.17 (Simplified Schur conditions for n “ 3).Let the characteristic polynomial be

P pλq “ λ3 ` a1λ2 ` a2λ ` a3

with a1 “ ´ trA, a2 “ sum of the subdeterminants and a3 “ ´ detA. Then thefollowing inequalities are necessary as well as sufficient for asymptotic stability

a) 1 ` a1 ` a2 ` a3 ą 0

b) 1 ´ a1 ` a2 ´ a3 ą 0

c) 1 ´ a2 ` a1a3 ´ a23 ą 0

d) a2 ą 3


(A remark by Gandolfo (p. 90): “Condition d) is redundant.” Counterremark byFranke: “No, it’s not, Gandolfo is wrong! For the example have a look at thehandout.”)

2.4. Application: The multiplier-accelerator model (MAM) (Samuelson/Hicks).Observation: The characteristic solutions of

yt ` a1yt´1 ` a2yt´2 “ 0

are constituted by the roots of

P pλq “ λ2 ` a1λ ` a2 “ 0

Samuelson/Hicks (1950).

Yt “ Ct ` It

Ct “ Ca ` cYt´1 pwith marginal propensity to consume (multiplier) 0 ă c ă 1qIt “ Ia ` βpYt´1 ´ Yt´2q

loooooooomoooooooon

accelerator

Superscript a stands for autonomous (i.e. is not changing). The feedback structureleads to

Yt ´ pc ` βqYt´1 ` βyt´2 ` constant term “ 0

Observation: detA “ a2 “ β and trA “ á1 “ c ` β. Therefore, the Schurconditions are

1 ` a1 ` a2 “ 1 ´ c ´ β ` β “ 1 ´ c ą 0 holds

1 ´ a1 ` a2 “ 1 ` c ` β ` β ą 0 holds

1 ´ a2 “ 1 ´ β!ą 0


Hence, the only relevant condition for the determination of stability is the 3rd one,AS ” β ă 1. There is cyclical behavior if ∆ “ ptrAq2 ´ 4 detA ă 0 and because oftaking the squareroot we get

β ą p1 ´ ?1 ´ cq2

or

β ă p1 ´ ?1 ´ cq2

Now we will represent these in a diagramm which gives us nice visualizations of(stable/unstable) monotonic/cyclic behavior.The function is given by β “ p1 ˘ ?

1 ´ cq2 with 0 ă c ă 1. As β increases the

0 0,4 0,8 1,2 1,6 2 2,4 2,8 3,2 3,6 4 4,4

-0,4

0,4

0,8

1,2

1,6

2

1

economy tends to explode, thus, β is destabilizing. The cyclical oscillations of themodel are either dampened (CS) or exploding (CU) and both isn’t persistent at all(not even in the case of |λ| “ 1 we can be sure without further information to verifypersistent oscillations). So the question is how to get persistent fluctuations, i.e.oscillations that neither explode or disappear (how to generate a business cycle).This is done in a very stylized way. There are two basic approaches and the firstone is also the common approach.

‚ Assume stability and add random shocks, which kick the system out ofequilibrium and fluctuations can set in again. If this is done skillfully oneends up with a shock-driven BC (mainstream RBC theory).

phenomenon corresponding causecyclical behavior intrinsic dynamics (i.e. from the deterministic core)persistent fluctuations exogenous (i.e. from random shocks)“period” T calibrate β, c

(“period” is in quotes because it is ragged, i.e. no nice shaped sine function)


‚ Assume instability and introduce non-linearities (such as boundaries: ceil-ing for investment function, f.e., floor). If this is skillfully done one endsup with an endogenous BC.

2.5. Some basic facts for rational expectations dynamics.General structure of RE models:

1) representative agents/decision units (HH, Firms): intertemporal optimiza-tion (over an infinite time horizon)

2) market clearing (budget equation)3) math. operations Ñ difference equation system: forward-looking compo-

nent (reduced form)4) Solve (with MATLAB/Dynare)5) Look at simulated model output (with MATLAB/Dynare)

Is this a good modelling approach? Ñ KISS: Keep It Sophisticatedly Simple

2.5.1. Solution approach to purely forward-looking models (NKB).

πt - inflation rate (zero in steady-state)yt - output gap (deviation (%) of aggregate output from trend output) [no

growth in the output gap]ρ - (long-run) equilibrium real interest rateit - nominal interest rate (set by the central bank)

The “microfoundations” plus RE gives the reduced form of the New KeynesianBaseline Model:

πt “ βEtπt`1 ` κyt [New Keynesian Phillips Curve (NKPC)]

yt “ Etyt`1 ´ 1

σpit ´ Etπt`1 ´ ρq [IS curve (investment/saving)]

it “ ρ ` ϕππt ` ϕyyt ` vt [Monetary Policy Rule (Taylor Rule)]

vt “ γvt´1 ` εt [Shock with εt „ Np0, σ2qswith β “ 0.99 (discount factor), κ “ combination of other parameters, σ “ in-tertemporal elasticity, ϕπ , ϕy “ policy paramter, 0 ă γ ă 1. Now let’s have a lookat the feedback

However, expectations are not given since they are part of the solution! They arenot determined by past expectations!Does the shock vt have an impact on the interest rate it? Conjecture: it Òyes - if the autocorrelation coefficient γ is sufficiently small (“small enough”)no - if the autocorrelation coefficient γ is sufficiently high (close to one)


(Problem: Transparency and clarity, because this is by no means obvious or intu-itive but simply the result of mathematical operations)The NKB can be reduced to

„

ytπt

“ A

„

Etyt`1

Etπt`1

´ Bvt

with

A “ c

„

σ 1 ´ βϕπ

σκ κ ` βpσ ` ϕyq

B “ c

„

1κ

c “ 1

σ ` ϕy ` κϕπ

First of all a little side note to start the lecture: as mentioned in the last couple oflectures we’re all using models mostly in an informal manner. In the lecture we’redealing with formal models which tend to bring discipline into economic reasoning.Such models can be distinguished via distinguishing their variables into

‚ predetermined (inherited from past or just lagged)‚ non-predetermined‚ shock variables

Determinacy :ðñ for each combination of the predetermined variables in t “0, and each bounded sequence of shocks εt pt “ 0, 1, . . .q D exactly one boundedsolution.Consider the deterministic core of

zt “ AEtzt`1 ´ Bvt

(all shock variables equal to zero rvt !“ 0 @ts). It follows that Etzt`1 “ zt`1 holdswhich leads to

zt “ Azt`1 ô zt`1 “ ztA´1

This system has an equilibrium z˚ “ 0. Since there are no predetermined variableswe are free to choose the initial condition. One obvious starting point would bez0 “ 0. But suppose z0 ‰ 0 such that corresponding tztu remains bounded:

1) solution is not unique2) D an EV λ of A´1 which satisfies |λ| ď 1 ùñ D an EV µ of A with |µ| “

ˇ

ˇ

1λ

ˇ

ˇ ě 1 (must not happen for determinacy)

Proof.

Bx “ µx

1

µx “ B´1x

ñ if µ is EV of B ô 1µis EV of B´1

Final conclusion: Determinacy of NKB ô both EVs of A must be strictly withinthe unit circle (|λ1,2| ă 1) to satisfy the requirement of a stable matrix. Using theSchur conditions we end up with the following conditional relationship:

κpϕπ ´ 1q ` p1 ´ βqϕy ą 0

If it holds determinacy prevails. The relationship is typically fullfilled if the famous“Taylor principle” ϕπ ą 1 occurs (the second term is nearly equal to zero because of


β « 0.99, thus, the first term must be positive and this obviously requires ϕπ ą 1).Determine the solution (“Law of motion”):

AEtxt`1 “ Bxt ` Dvt (13)

vt`1 “ Nvt ` εt

with x P Rn, A,B P Rnˆn, v, ε P Rm pm ď nq (at least as much state variables asshock variables), N P Rmˆm, D P Rnˆm, Etεt “ 0Assumption: B is invertible and N is stable (with N unstable possibly the shockswill explode)

Proposition 2.18. Suppose determinacy prevails for (13). Then there exists auniquely determined matrix S P Rnˆm such that a solution of (13) is given by

xt “ Svt (14)

vt “ Nvt´1 ` εt´1

(Note that (14) is a system determined just by shocks, i.e. shock driven.)Observation: Dynamics of xt are exclusively governed by shocks; no intrinsic dy-namics

Proof. Determine S by the method of undetermined coefficients. Lead (14) by oneperiod

xt`1 “ Svt`1

take expectations, conditional on t

Etxt`1 “ EtSvt`1(13)“ EtSNvt ` Etzt “ SNvt

Substitute this (14) in (13)

ASNvt “ BSvt ` Dvt

This is to hold @t. Hence, S has to satisfy

ASN “ BS ` D p˚qSolve p˚q for S:

vecpSq “ “´pI b Bq ` pN 1 b Aq‰´1vecpDq

This leads to

yt “ ´p1 ´ βγqΛvtπt “ ´κΛvt

it “ ρ ` CΛvt

with

Λ :“ 1

p1 ´ βγqrσp1 ´ γq ` ϕys ` κpϕπ ´ γq ą 0

and

C :“ σp1 ´ γqp1 ´ βγq ´ κγ ž 0

The sign of C depends on the persistence in the shock process. For systems whichare of slightly higher order such an explicit solution can’t be calculated, however anumerical analysis can be done.


2.5.2. Solution approach to univariate hybrid models. Additionally to forward-lookingterms (Etxt`1) hybrid models typically include also lagged values (xt´1).Example: [1]

kt “ a1Etkt`1 ` a2kt´1 ` bzt

zt “ ρzt´1 ` εt´1 pwith Etεt “ 0; |ρ| ă 1q (15)

In order to detect determinacy the task is to determine the deterministic core bydropping all random terms. It follows that

kt`1 ` c1kt ` c2kt´1 “ 0

and we have to compute the EVs λ1,2. There are 3 possible cases which can occur:

1) |λ1| ă 1 ă |λ2| - this situation refers to saddle point stability and is bothsufficient as well as necessary condition for determinacy.

2) |λ1|, |λ2| ă 1 - convergence (boundedness) for all initial conditions (nodeterminacy because uniqueness is violated)

3) |λ1|, |λ2| ą 1 - divergence for all initial conditions (no determinacy becauseboundedness is violated)

Thus, the claim can be made that determinacy is achieved if |a1 ` a2| ă 1 holds.

Proposition 2.19. Suppose determinacy prevails for (15). Then the solution of(15) is uniquely determined by

kt “ ϕkkt´1 ` ϕz

ρzt (16)

where with respect to

ϕ1,2 “ 1

2a1t1 ˘ ?

1 ´ 4a1a2uϕk is the unique number ϕ1 or ϕ2 that is less than unity in modulus (which existsby virtue of determinacy), and

ϕz “ bρ

1 ´ a1pρ ` ϕkq(Note that there is a special case with a2 “ 0 which leads to the purely forward-looking model. Its solution is compatible with that of the previous lecture/subsection.)

Proof. (Method of undetermined coefficients (MoUC))Write (16) as

kt “ ϕkkt´1 ` ϕzzt´1 ` ϕz

ρεt

Lead it by one period and take conditional expectations

Etkt`1 “ ϕkkt ` ϕzzt ` 0

Substitute in (15)

kt “ a1pϕkkt ` ϕzztq ` a2kt´1 ` bzt

and collect terms such that

p1 ´ a1ϕkqkt “ a2kt´1 ` pa1ϕz ` bqpρzt´1 ` εtqkt “ a2

1 ´ a1ϕk

kt´1 ` pa1ϕz ` bqρ1 ´ a1ϕk

zt´1 ` pa1ϕz ` bq1 ´ a1ϕk

εt

ϕk “ a2

1 ´ a1ϕk

and ϕz “ pa1ϕz ` bqρ1 ´ a1ϕk


(Note that the first (second) equation/relation is a quadratic (linear) equation inϕk (in ϕz).)

2.5.3. Solution approach to general hybrid models. According to Dynare: It solves

EttAyt`1 ` Byt ` Cyt´1 ` Dutu “ 0 (17)

ut “ Nut´1 ` εt (18)

with y P Rn, u P Rm, Etεt “ 0

Proposition 2.20. If determinacy prevails for (17) and (18) then there exists(unique) matrices Ω P Rnˆn,Γ P Rnˆm such that for y´1, u0 given, the solutiontytu8

t“0 is determined by (19) and Ω is stable (leads to convergence and, hence, issufficient condition for determinacy).

yt “ Ωyt´1 ` Γut (19)

Determination of determinacy:

predetermined non-predeterminedyi,t´1 all noneAssumption: yi,t n ´ q q pq ď nq

(Additionally assumption: A is invertible (if not, respecify dating of the variables).)Solve for the deterministic core of (17) for yt`1

yt`1 “ Á´1Byt ´ A´1Cyt´1

ùñ„

yt`1

yt

“„Á´1B Á´1C

I 0

„

ytyt´1

ùñ„

yt`1

yt

“ G

„

ytyt´1

Recipe: Blanchard-Kahn conditions (generalized saddle-point stability)

Determinacy prevails if (exactly) q EVs of G are unstable (outside the unit circle).

Proof. (MoUC)Lead (19) by one period and take conditional expectations

yt`1 “ Ωyt ` Γut`1

(19),(18)“ Ω rΩyt´1 ` Γuts ` ΓNut ` Γεt

“ Ω2yt´1 ` pΩΓ ` ΓNqut ` Γεt

Substitute this and (19) in (17)

EttAΩ2yt´1 ` ApΩΓ ` ΓNqut ` AΓεt`1 ` BΩyt´1 ` BΓutlooooooooomooooooooon

yt

`Cyt´1 ` Dutu “ 0

rAΩ2 ` BΩ ` Csyt´1 ` 0 ` rAΩΓ ` AΓN ` BΓ ` Dsut “ 0

This relationship holds in general for all combinations of yt ´ 1 and ut if

AΩ2 ` BΩ ` C “ 0 (20)

AΩΓ ` AΓN ` BΓ ` D “ 0 (21)

Thus, one has to solve for Ω and Γ. To get Γ consider (21) and define for sim-plification F :“ AΩ ` B which transforms (21) into FΓ ` AΓN ` D “ 0 (notethat in the special case of N “ 0 (white noise) the solution takes simply the formΓ “ ´F´1D). In general the solution takes the form

vecpΓq “ ´ “pI b F q ` pN 1 b Aq‰´1vecpDq


and can have many solutions, which might not be stable. In order to solve (20)for Ω we have to apply the (Generalized) Schur (QZ) Decomposition or an iterativealgorithm which gives an approximate solution:

AX2 ` BX ` C “ 0

pAX ` BqX “ Ć

X “ ´pAX ` Bq´1C

Since the last equation cannot be solved directly, suppose there exist a k such that

Xk`1 “ ´pAXk ` Bq´1C

This is the initial equation of the iteration process. The following can be seen as arecipe to solve for Ω

‚ Choose an initial starting matrix X0 “sufficiently” close to the true solutionin a sense that convergence is assured

‚ Set X0 “ αI with 0 ă α ă 1 as initial value, propose α “ 0.8 (α should beless than one to begin the iteration with a stable matrix X0 which leads toa higher probability of obtaining an also stable matrix Ω)

‚ Iterate untilmaxij

|xk`1,ij ´ xk,ij | ă ε

for some (small) tolerance level ε ą 0 (ε as precision parameter), then putthe last matrix of the iteration process X˚ equal to xk`1

‚ Check whether X˚ satisfies (20) approximately‚ Check whether all EVs of X˚ are less than 1 in modulus/absolute value‚ If this holds, put Ω “ X˚

3. Ordinary differential equation (ODE)

3.1. The mother of all differential equations. Let

9x “ ax (with xp0q “ x0 given)

be the derivative of function x at time t which is equal to ax, a times the value ofthr function in time t.Solution: xptq “ eatx0

1) initial condition xp0q “ 1x0 (is true)2) 9xptq “ d

dteatx0 “ aeatx0 “ axptq

Example: Cumulative saving in the continuous-time limit

S - saving per year (const.)r - interest rate p.a.

W ptq - wealth at time t

W0 - initial wealth

a) saving and interest payments (reinvested) every year

W pt ` 1q “ W ptq ` rW ptq ` S with W p0q “ W0

b) every h years (h-economy)

W pt ` hq “ W ptq ` hrW ptq ` hS

ô W pt ` hq ´ W ptqh

“ rW ptq ` S

c) continuous time: h Ñ 0

9W “ rW ` S


Solution: W ptq “ cert ´ Srwhere c “ S

r` W0

Proof.The same procedure as above...

1) initial condition W p0q “ Sr

` W0 ¨ 1 ´ Sr

“ W0 (is true)2)

9W ptq “ rcert ´ S ` S

“ r

„

cert ´ S

r

` S

“ rW ptq ` S

3.2. Types of continuous-time systems.Let y P R the second order differential equation is given by

:y ` a1 9y ` a2y “ 0

More generally

(1) Inhomogenous linear n-th order differential equation

ypnq ` a1ypn´1q ` . . . ` an´1 9y ` any ` y “ 0

with y, y P R

(2) Homogenous linear n ˆ n differential equation system

9x “ Ax

with x P Rn, A P Rnˆn

(3) Inhomogenous linear n ˆ n differential equation system

9x “ Ax ` x

(4) General non-linear, autonomous differential equation system

9x “ fpxqwith f : Rn Ñ Rn (any generally non-linear mapping)

(5) General non-linear differential equation system

9xptq “ fpt, xptqqwith f : Rn`1 Ñ Rn (non-autonomous). Example: 9xptq “ gpxptqqà ¨sinptq

All of these equations/systems above are ODE. Example for non-ordinary differen-tial equation system: 9xptq “ axptq ` bxpt ´ T q (new quality: 8 many EVs!)NK:

xt “ αEtxt`1 ` p1 ´ αqxt´1 ` . . .

with α P r0, 1sa) Et “ Et “ math. expectation = rational expectation (RE)

b) Et “ ϕtEb1t ` p1´ϕtqEb2

t with b « bounded rationality and two forcastingrulesYt endogenously increases with forecasting succes of rule 1 (invented by P.DeGrauwe (2010))

– several successors– might gain momentum

NK3, hybrid:Taylor rule: rt “ . . . πt . . . yt

Byt “ ÁEtyt`1 ´ Cyt´1 ´ Dut

with B,A,C P R2ˆ2


‚ θ P Rp (“deep” parameters):

θmodelÝÝÝÝÑ Apθq, Cpθq (20)ÝÝÑ Ω “ Ωpθq (solution matrix)

and

Rp Ñ R

4ˆ4 Ñ R2ˆ2

‚ Suppose: θ0 is “true” parameter vector Ñ Ω0 “ Ωpθ0qObservation: p ą 4 ñ D (infinitely) many θ : Ωpθq “ Ω0 ñ no estimationcan identify θ0

Solution:

1.) Does a solution exist?2.) Is it unique?3.) Can it be explicitly determined? Ñ Yes, for linear systems!4.) rë if nos Can it be numerically solved, simulated?

“Euler-one-step-procedure” (simulation):

9xptq “ fpxptqqxpt ` hq ´ xptq

h“ fpxptqq

xpt ` hq “ xptq ` h ¨ fpxptqqAdvantage Disadvantagepractically, good enough D better approximation methodsoften economic interpretation (like Runge-Kutta-approximation)

Proposition 3.1. Transform (inhomogeneous) equation (1) into homogeneousequation (1).

Proposition 3.2. A homogeneous equation (1) can be transformed into an n ˆ n

differential equation system (2).

Proposition 3.3. Technical subleties apart, (2) can be transformed into (1).

Proposition 3.4. If (3) has a unique equilibrium x˚, it can be reduced to (2).More precisely, if xp¨q is a solution of (3), zp¨q “ xp¨q ´ x˚ satisfies 9z “ Az.

Agenda: study 9x “ Ax (HDE)

3.3. Homogeneous linear differential equation systems.

Proposition 3.5 (Superposition Theorem). Let xp¨q, yp¨q be two solutions of(HDE) and zp¨q “ axp¨q ` byp¨q, for some a, b P R. Then also zp¨q is asolutionof (HDE).

Remark: See appropriate DT remark.Significance of Proposition 5:

‚ find “nice” (characteristic) solutions of HDE‚ verify that any arbitrary solution of (HDE) is a suitable linear combinationof these “nice” solutions


Recall: Proposition 6 (DT): xt “ Atx0. For that, recall: ex “ 1 ` x ` x2

2! ` x3

3! `x4

4! ` . . .. This motivates eA :“ I ` A ` A2

2! ` A3

3! ` A4

4! . . . ùñ converges (!) to amatrix @A.

Proposition 3.6. Given any initial condition x0 P R, the (unique) solution of(HDE) is given by

xptq “ eAtx0

Proposition 3.7. If λ P C is an EV of A with eigenvector v P Cn (i.e. Av “ λv),

then, starting from x0 “ v,

xptq “ eλtx0 (22)

Proof.

1.) t “ 0 : xp0q “ 1 ¨ x0

2.) 9xptq “ λeλtx0 “ eλtpλvq “ eλtAv “ Aeλtv “ Axptq

CT (xptq “ eλtx0) DT (λtv)Jumps are not possible jumps are possible (jumps of points)

λ ą 0 ñ divergence λtv ñ |λ| £ 1λ ă 0 ñ convergence (towards zero or equil. point)

with λ P R and for complex EV λ “ α ˘ iβ

ñ eα˘iβ “ eα pcospβq ˘ i sinpβqq pα ˘ iβ “ r pcospθq ˘ i sinpθqqq

Proposition 3.8 (see DT for full text). [. . . ] Then for any c1, c2 P R

xptq “ eαt rcospβtqrc1w ` c2zs ` sinpβtqrc2w ´ c1zss (23)

is a (real-valued) solution of (HDE).

Note: α here is responsible for con-/divergence motion

ñ α “#

real componentpλq ă 0 ñ spiralling inward

real componentpλq ą 0 ñ spiralling outward

Period: sinp0q “ sinpβT q “ sinp2πq ñ T “ 2πβ

“ 2πImag.pλq

Proposition 3.9 (normal/regular case (=distinct EV)). Suppose A P Rnˆn hasn distinct EVs. . . [see DT]

Proposition 3.10. Let λ “ α ˘ iβ be an EV (real (β “ 0) or complex β ‰ 0)of A P Rnˆn with “multiplicity” mpď nq. Then the solution xp¨q to which this EVgives rise (analogously to Prop. 7 and 8) is given by

xptq “ eαt

»

—

—

—

–

p1ptqp2ptq...

pnptq

fi

ffi

ffi

ffi

fl

(24)

where p1p¨q, . . . , pnp¨q are polynomials (of complex numbers) in t of order less orequal to m ´ 1.


Remark 1/2: See DT«

eαtp`q

pαqff8

´8

Most important observation:

α “#

real componentpλq ă 0 ñ convergence to zero vector

real componentpλq ą 0 ñ divergence

Proposition 3.11. see DT

3.4. Stability conditions. Consider type (4) of the continous-time systems 9x “ypxq, f : Rn Ñ R⋉ and Jacobian at an equilibrium point x˚

J “

»

—

–

f11 f12 . . . f1n...

.... . .

...fn1 fn2 . . . fnn

fi

ffi

flwith fij :“ Bfipx˚q

Bxj

pi, j “ 1, . . . , nq

Linearized system of (4): 9x “ Jpx ´ x˚q p2aq

Proposition 3.12. Suppose all EV of J (of x˚) are different from zero. Then thefollowing holds:

(a) The equilibrium x˚ is locally unique(b) Locally around x˚, the trajectories of (4) are qualitatively the same as those

of (2a)

In particular:

‚ (2a) is LAS ñ (4) is LAS‚ (2a) is repelling |Rightarrow (4) is repelling‚ (2a) is a saddelpoint ñ (4) is a saddlepoint

Consider 9x “ Ax,A P R2ˆ2, n “ 2 and compute the EVs:

‚ trpAq “ a11 ` a22‚ detpAq “ a11a22 ´ a12a21‚ ∆ “ ptrpAqq2 ´ 4 detpAq‚ ñ λ1,2 “ 1

2

“

trpAq ˘ ?∆

‰

Now there are 3 possible cases:

Case 1: trpAq ‰ 0 and detpAq ď 0 ñ ?∆ ě | trpAq| ą 0 ñ λ1 ď 0 ď λ2 ñ

detpAq ą 0

is necessary condition for stability (LAS); [x˚ is saddle-point ô detpAq ă0].

Case 2: trpAq ě 0 ñ one EV of A is real and non-negative (if ∆ ě 0) or EVs arecomplex (∆ ă 0) but have non-negative real parts (α ě 0)

trpAq ă 0

is necessary condition for stability.Case 3: trpAq ă 0, detpAq ą 0: distinguish between

a) ∆ ă 0 ñ λ1,2 P C with real component λ1,2 “ trpAq2 ă 0 ñ stability

‘

b) ∆ “ 0 ñ λ1,2 P R with λ1,2 “ trpAq2 ă 0 ñ stability

‘

c) ∆ ą 0 ñ ?∆ ă aptrpAqq2 “ | trpAq| ñ λ1,2 ă 0 ñ stability

‘


Proposition 3.13. A sufficient as well as necessary condition for A P R2ˆ2 tohave two EVs with real parts is

trpAq ă 0, detpAq ą 0

tr(A)

MU

CUCS

MS

det(A)

∆=0

saddle-points

∆ ď 0 ô ptrpAqq2 ´ 4 detpAq ď 0 ô detpAq ě 14 ptrpAqq2

3.5. Stability conditions (for higher-dim. systems).

Theorem 3.1.

‚ RCpλq ă 0 @ EVs Ñ x˚ LAS‚ RCpλq ă 0 for at least one EV Ñ x˚ unstable‚ RCpλq “ 0 for at least one EV Ñ no conclusion without further information

Theorem 3.2. Coefficients in the characteristic polynomial

a1 “ ´ trpJq . . . an “ p´1qn detpJq

Theorem 3.3.

trpJq “n

ÿ

i“1

λi; detpJq “n

ź

i“1

Consequence 1: Necessary for LAS

a) trpJq ă 0b) p´1qn detpJq ą 0

Consequence 2: On the LHS of the critical value: Rcpλq ă 0 @λ; RHS of the critical

value: Dλ : Rcpλq ą 0 (Assumption: p´1qn detpJpµHqq ą 0)

Theorem 3.4 (General Routh-Hurwitz conditions). LAS iff

∆1 ą 0,∆2 ą 0, . . . ,∆n ą 0

(iff = necessary & sufficient = ô)

‚ complicated (very)‚ some inequalities are implied by others‚ n “ 2, n “ 3, n “ 4 can be further simplified


Rc

Imag.

µ=µH

µH

(critical value)

µ

Theorem 3.5 (Routh-Hurwitz for 2D).

‚ LAS iff trpJq ă 0, detpJq ą 0‚ EVs are purely imaginary ô trpJq “ 0, detpJq ą 0

Theorem 3.6 (Routh-Hurwitz for 3D). Put

‚ a1 :“ ´ trpJq‚ a2 :“sum of the principal 2nd-order minors of J

“∣

∣

∣

∣

f11 f12f21 f22

∣

∣

∣

∣

`∣

∣

∣

∣

f11 f13f31 f33

∣

∣

∣

∣

`∣

∣

∣

∣

f22 f23f32 f33

∣

∣

∣

∣

with J “»

–

f11 f12 f13f21 f22 f23f31 f23 f33

fi

fl

‚ a3 :“ ´ detpJq‚ b :“ a1a2 ´ a3

Then x˚ is LAS ô a1 ą 0, a3 ą 0, b ą 0

Remark:

1) a1 ą 0, a3 ą 0, b ą 0 ñ a2 ą 02) a2 ą 0, a3 ą 0, b ą 0 ñ a1 ą 0

Theorem 3.7 (Routh-Hurwitz for 4D). Define a1, a2, a3, a4, b :“ a1a2a3 á21a4 á23. LAS ô ak ą 0 for k “ 1, 2, 3, 4; b ą 0

Application in 3D: (The Lorenz equation/butterfly effect)

(1) 9x “ σp´x ` yq(2) 9y “ ρx ´ y ´ xz

(3) 9z “ xy ´ βz

(Assumptions: σ, ρ, β ą 0, σ ą 1 ` β)Equilibria:

a) x0 “ y0 “ z0 “ 0 (trivial solution)

b) additional 9x “ 0 ñ x “ y and 9z “ 0 ñ x2 ´ βz “ 0 ñ z “ x2

β

substitute in (2): 9y “ 0 ñ pρ ´ 1qx ´ x3

β“ σ

x‰0ùñ βpρ ´ 1q “ x2 ρą1ùñ x1,2 “y1,2 “ ˘a

βpρ ´ 1q Ñ z1,2 “ ρ ´ 1

Jacobina at E0, E1, E2

JpEkq “¨

˝

´σ σ 0ρ ´ zk ´1 ´xk

yk xk ´β

˛

‚


Stability at “inner equilibrium” E0:

JpE0q “¨

˝

´σ σ 0ρ ´1 00 0 ´β

˛

‚

The 3rd column leads to ´β “ λ1 and λ2,3 are given by the EVs of J2 “ˆ´σ σ

ρ ´1

˙

.

Applying Routh-Hurwitz conditions gives us trpJzq “ ´p1 ` σq ă 0 and

detpJzq “ σp1 ´ ρq ñ#

E0 is LAS if ρ ă 1 pbecause detpJzq ą 0qE0 is unstable if ρ ą 1 pbecause detpJzq ă 0q

Check this via implementing Routh-Hurwitz (3D):

‚ a1 “ ´ trpJq “ 1 ` σ ` β ą 0‚ a2 “ βpσ ` ρq ą 0‚ a3 “ 2σβpρ ´ 1q ą 0

‚ b “ β

»

—

–p1 ` β ´ σqlooooomooooon

ă0 pσą1`βq

ρ ` σp3 ` σ ` βq

fi

ffi

flž 0 (depends on ρ Ñ solve for ρ)

Define

ρH “ σp3 ` σ ` βqσ ´ β ´ 1loooomoooon

ą0

(Hopf Bifurcation)

Thus,

E1,2 “#

LAS if b ą 0 ô 1 ă ρ ă ρH

unstable if ρ ą ρH


References

[1] Carceles-Poveda, E., Giannitsarou, C. (2007), Adaptive learning in practice, Journal ofEconomic Dynamics & Control, 31, pp. 2659-2697.

[2] Gandolfo, G. (1997), Economic Dynamics. Berlin: Springer.[3] Luetkepohl, H. (2007), New Introduction to Multiple Time Series Analysis. Berlin: Springer.[4] Tu, P.N.V. (1994), Dynamical Systems: An Introduction with Applications in Economics and

Biology. Berlin: Springer (2nd revised and enlarged edition).[5] Woods, J.E. (1978), Mathematical Economics. London, New York: Longman (Chapter 4).

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LECTURE NOTES ON ADVANCED …mathematical-methods.wikispaces.com/file/view/MM.pdf2 ADVANCED MACROECONOMICS I - WS 2011/2012 1. ... ñ solution is tied down to the present ñ solution