004: Macroeconomic Theory

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004: Macroeconomic Theory

Lecture 7

Mausumi Das

Lecture Notes, DSE

August 8, 2014

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First order Autonomous Di¤erence Equation: Steady State(s)

As we have discussed in the previous lecture, the steady state of a

…rst order autonomous di¤erence equation of the form xt+1 =f(xt;α)is de…ned as

¯

x : ¯x=f(x;¯ α).

Depending on the nature of thef function, solution to the above equation may not exists; may be unique or may even be multiple in number.

Given that we are interested in economic variables, we shall often restrict our analysis to steady state values that arenon-negative.

An alternative representation of the steady state is given by the solution to the following two simultaneous equations:

x_t+1 =f(x_t;α) (i) x_t₊₁=x_t (ii)

Das (Lecture Notes, DSE) Macro August 8, 2014 2 / 15

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First Order Autonmous Di¤erence Equation: Steady State(s) (Contd.)

It is often convenient to diagrammatically characterise the steady state(s) in the xt-xt+1 plane by simultaneously plotting (i) and (ii) and identifying the intersection point(s).

Such a diagram is called thePhase Diagram. The advantage of the phase diagramme technique is that not only does it allow us to identify the possible steady states, it also allows us to characterise the behaviour of the state variable outside the steady state and thus analyse the stability of various steady states.

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First Order Autonmous Di¤erence Equation: Steady State(s) (Contd.)

Phase Diagram: An Illustration (A Unique Steady State)

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First Order Autonmous Di¤erence Equation: Steady State(s) (Contd.)

Phase Diagram: An Illustration (No Steady State)

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First Order Autonmous Di¤erence Equation: Steady State(s) & Stability

Phase Diagram: An Illustration (Multiple Steady States)

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First Order Autonmous Di¤erence Equation: Stability

Consider an automous …rst order di¤erence equation of the form:

xt+1 =f(xt;α).

Let x¯ be a steady state of this dynamical system such that

¯

x =f(x;¯ α).

Stability: The steady statex¯ is said to bestable if for every e>0 there exists a δ 2(0,e] such that

j^x^s ^x^¯j<δ) j^x^t ^x^¯j<e for all t =^s^. Asymptotic Stability: The steady state x¯ is said to be

asymptotically stable if for every e>0 there exists a δ 2(0,e) such that

(i) For all t = ^s, j^x^s ^x^¯j<δ) j^x^t ^x^¯j<e (ii) Ast ! ∞, j^x^t ^x^¯j !⁰

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First Order Autonmous Di¤erence Equation:

Stability(Contd.)

A Diagrammatic Illustration of Stable Steady States vis-a-vis Asymptotically Stable Steady States:

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First Order Autonmous Di¤erence Equation: Stability (Contd.)

Asymptotic Stability: Global vis-a-vis Local

The largest neighbourhood form which from which in entering trajectory converges asymptotically to the steady statex¯ is called the basin of attraction ofx.¯

If this region is the entire state space < then the steady statex¯ is called a globally stable.

A less abstract (and more workable) de…nition of Global vis-a-vis Local Asymptotic Stability (in terms of initial value):

A steady statex¯ is globaly asymptotically stable if

tlim!∞xt !^x^¯ ^{for all}^x0 2 <^.

A steady statex¯ is locally asymptotically stable if we can …nd an e>0 such that

tlim!∞x_t !^x^¯ ^{for all}^x0 for which j^x0 x¯j<e.

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Steady State of the First Order Linear & Autonmous Di¤erence Equation:

Let us now go back to the …rst order linear and autonomous di¤erence equation of the form, discussed in the previous lecture:

xt+1 =axt+b.

We have seen that its general solution is given by:

xt = 8<

:

Ca^t + ^b

1 a ifa6=1 C +bt ifa=1

We have also seen that whena6=1, the uniques steady state of the dynamical system is given by:

¯ x = ^b

1 a.

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Steady State of the First Order Linear & Autonmous Di¤erence Equation (Contd.):

Notice that whena=1, this system can have multiple steady states provided b =0.

In fact when a=1 andb =0, the di¤erence equation is given by:

x_t+1 =x_t.

The system obviously has multiple steady states. Every point in the state space< is a steady state to this system!

Having identi…ed all the possible steady states of this linear and autonomous system, we would now like to examine the stability property of these steady states.

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Stability of the First Order Linear & Autonmous Di¤erence Equation:

There are two ways to analyse the stability property of a dyanamic system:

We can directly look at the solution (if we can solve the system) and then analyse the asymptotic property of the solution;

We can use the Phase Diagram approach to diagramatically

characterize the steady state and identify the behaviour of the system when the system is not at steady state.

Here we take the …rst approach. The Phase Diagram approach will be discussed in the next class.

Case A:a6=1

Dynamical System:

x_t+1 =axt +b Solution:

xt =Ca^t+x;¯ C an arbitrary constant.

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Stability of the First Order Linear & Autonmous Di¤erence Equation (Contd.):

Case A1: 0<a<1.

The system converges monotonically to the steady statex, which is the¯ only steady state of the equation.

This happens independent of the value ofC (i.e., the initial/boundary conditions). Hence the steady state isglobally aymptotically stable.

Case A2: a>1.

The system diverges monotonically away from the steady statex.¯ Hence the steady state isunstable.

Case A3: 1<a<0.

The …rst term in the solution changes sign in alternative time periods;

so the solution ‡uctuates around the steady statex.¯

Nonetheless the magnitude of the ‡unctions decreases over time (damped oscillations)

Hence the steady state isglobally aymptotically stable.

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Stability of the First Order Linear & Autonmous Di¤erence Equation (Contd.):

Case A4: a< 1.

The …rst term in the solution again changes sign in alternative time periods; so the solution ‡uctuates around the steady statex.¯ But now the magnitude of the ‡unctions increases over time (explosive oscillations)

Hence the steady state isunstable.

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Stability of the First Order Linear & Autonmous Di¤erence Equation (Contd.):

Case B:a=1;b =0 Dynamical System:

x_t₊₁ =x_t Solution:

x_t =C; C an arbitrary constant.

As we argued earlier, the system has multiple steady states. Any point in the state space is a steady state point for this system.

Question: Can you comment on the stability property of this system?