004: Macroeconomic Theory

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004: Macroeconomic Theory

Lecture 8

Mausumi Das

Lecture Notes, DSE

August 12, 2014

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First Order Non-Linear Autonmous Di¤erence Equation:

Stability

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

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

¯

x : ¯x=f(x;¯ α).

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

As discussed in the last class, even when we are not able to solve the above equation, if enough information about the f function is given then we can diagramatically characterise the steady state in terms of aPhase Diagram.

In today’s class we shall use the phase diagram to analyse the stability property of the dynamical system.

Das (Lecture Notes, DSE) Macro August 12, 2014 2 / 16

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First Order Autonmous Non-Linear Di¤erence Equation:

Stability (Contd.)

As we have seen before, 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) xt+1=xt (ii)

Drawing thePhase Diagram entails simultaneously plotting of (i) and (ii) in thext-xt+1 plane.

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Using Phase Diagram to analyse Stability:

Phase Diagram: An Illustration (A Steady State which is Globally Asymptotically Stable)

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Using Phase Diagram to analyse Stability: (Contd.)

Phase Diagram: An Illustration (A Steady State which is Unstable)

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Using Phase Diagram to analyse Stability (Contd.):

Phase Diagram: An Illustration (Multiple Steady States - some

‘locally’stable; some ‘locally’unstable)

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Stability of First Order Autonmous Non-Linear Di¤erence Equation: Linearization

The Phase Diagram technique helps us a to do a qualitativeanalysis of stability of the dynamical system.

Often if the di¤erence equation is non-linear, we compliment the phase diagram analysis with another technique called Linearization (around a steady state).

The basic idea is as follows:

Given any non-linear function of a single variable,f(x), and given any speci…c value ofx which lies in the domain off, sayx,˜ we can always fnd alinear approximationof the non-linear function in a close neighbourhood ofx,˜ by applying the Taylor’s Expansion Rule.

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Stability of First Order Autonmous Non-Linear Di¤erence Equation: Linearization (Contd.)

Taylor’s Expansion Rule states that iff(x)is in…nitely di¤erentiable atx˜ then:

f(x) = f(x˜) + ¹

1!f⁰(x˜)(x x˜) + ¹

2!f⁰⁰(x˜)(x x˜)²+ 1

3!f⁰⁰⁰(x˜)(x x˜)³+...

Now taking the …rst (linear) approximation:

f(x) =f(x˜) +f⁰(x˜)(x x˜).

Notice that the linear approximation allows us to represent the nonlinear functionf(x)by a linear function in a close neighbourhood of x, where the linear function has the following form:˜

f(x) =Ax +B;

whereA f⁰(x˜) andB f(x˜) f⁰(x˜)x˜ are both constants (since they are functional values evaluated at a speci…c point).

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Stability of First Order Autonmous Non-Linear Di¤erence Equation: Linearization (Contd.)

It is now easy to see how we can combine the linearization technique with the phase diagram technique to make a more rigorous statement about stability (albeit local stability).

Conside a …rst order autonomous di¤erence equation of the form x_t+1 =f(x_t) (ignoring the parameters for the moment).

Suppose we have identi…ed from the phase diagram (or from direct algebric calculation), that this dynamical system has three steady states, as shown in an earlier example:

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Stability of First Order Autonmous Non-Linear Di¤erence Equation: Linearization (Contd.)

We can now linearly approximate the f(xt) function at each of these steady states and approximate the non-linear dynamical system by a linear dynamical system in the neighbourhood of the chosen steady state.

Since we know the precise condition for stability (and instability) of a linear system we can use that information to comment about the (local) stability property of this ‘linearized’system.

For example, let us linearize f(x_t)around the second steady state x¯₂. Then we have

xt+1 =Axt+B,

whereA f⁰(x¯2)andB f(x¯2) f⁰(x¯2)x¯2 =x¯2 f⁰(x¯2)x¯2 (since

¯

x₂ itself is a steady state)^x^¯² =f(x¯₂)).

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Stability of First Order Autonmous Non-Linear Di¤erence Equation: Linearization (Contd.)

We know that the general solution to this ‘linearized’di¤erence equation is given by:

C(A)^t + ^B

1 A =C(A)^t+x¯₂.

We also know that stability requires j^Aj<1, i.e, j^f⁰(x¯₂)j<1.

Since the f(x_t)curve is intersecting the 45^o line from below at this point, it must be the case that at this pointf⁰(x¯2)>1.

Thus this steady state point must be locallyunstable.

This is precisely the conclusion that we reached by drawing the arrows in the Phase Diagram!

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First Order Linear Non-Autonmous Di¤erence Equation:

Solution & Stability

So far we have focussed our attention on autonomous di¤erence equations.

We have argued earlier that the concept of steady state is not de…ned for non-autonomous systems.

However, we may still be interested to know whether the system is asymptotically stable, in the sense whether it converges tosome constant value in the long run.

Consider a …rst order non-autonomous linear di¤erence equation of the form:

x_t₊₁=ax_t+b_t,

whereb_t is the non-autonomous term which is directly a function of time.

Examples: b_t =B(β)^t;b_t =Bt;b_t =exp(βt).

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First Order Linear Non-Autonmous Di¤erence Equation:

Backward Solution

If we knew the initial valuex¯₀,we could interate it backward such that x₁ = ax₀+b₀

x₂ = ax₁+b₁ =a(ax₀+b₀) +b₁ =a²x₀+ab₀+b₁ x₃ = ax₂+b₂ =a³x₀+a²b₀+ab₁+b₂

...

xt = a^tx0+a^t ¹b0+a^t ²b1+...+abt 2+bt 1

= a^tx₀+

t 1

i

∑

=0

aⁱb_t ₁ _i.

The corresponding general solution (which is called a Backward Solution) is given by:

t 1

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Backward Solution: Stability

The backward solution derived above pre-supposes existence on an initial condition. In fact di¤erent initial conditions generate di¤erent values for the arbitary constant C.

However, irrespective of the initial value (i.e., irrespective of the value of C),the system will be globally asymptotically stable if as t !^∞^,^x^t !^{x, where}^¯ ^x^¯ is some well-de…ned constant (but nota steady state).

A set of su¢ cient conditions for that are given by:

1 j^aj<1

2 j^btj5^B ^{for all}^t,^where ^B is a …nite constant.

To get a sketch of a proof as to why this is indeed su¢ cient, note that

∑

∞ i=0

aⁱb_{t i} 5

∑

^∞

i=0

aⁱB =B

∑

∞ i=0

aⁱ.Now proceed from here using the

condition thatj^aj<1 to show that

∑

∞ i=0

aⁱb_{t i} would then converge to a …nite value.

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First Order Linear Non-Autonmous Di¤erence Equation:

Forward Solution

Sometimes, for some state variables the initial value is not given, but a terminal condition is given instead.

Since for these variables the initial value is not pre-determined, i.e., the initial value is free to jump anywhere within the state space (as long as the terminal condition is satis…ed), such variables are called

‘jump variables’.

If we knew that the system will approach some terminal value as t !∞,then we could rewrite the di¤erence equation to express the current value of the variable in terms of its future value as follows:

xt = ¹ axt+1

1 abt

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Forward Solution: Stability

Now interating forward,

xt = ¹ axt+1

1 abt

= ¹ a

1 ax_t+2

1 ab_t+1

1

ab_t = ¹ a

2

x_t+2

1 a

2

b_t+1

1 ab_t

= ¹

a

3

x_t₊₃ 1 a

3

b_t₊₂ 1 a

2

b_t₊₁ 1 ab_t ...

= ¹

a

n

x_t₊_n 1 a

∑

n i=0

1 a

i

b_t₊_i.

We then allown to appoach∞and apply the given terminal

condition to arrive at a precise solution of the system. Such a solution is called the forward solution to the …rst order linear &

non-autonomous di¤erence equation speci…ed earlier.

When is this solution stable? (To be discussed in the next class).