Macro3Discrete Time Dynamic System a Technical Preparation(宏观经济学-南开大学,龚刚)

II. The Discrete Time Dynamic System

Let Xt, Yt, …, Zt are respectively the different variables at period t. Then their determination might be described by the following dynamic system, which is in discrete time:
No Converging to the Steady State
X Y
X* Y*
t
X*
X
III. The Solution Path and the Steady State

The steady state (continued)
There is no warranty that the steady state is unique (multiple steady states could occur) There is also no warranty that the steady state even exists (it could be complex). Both of the above are more likely to occur in a nonlinear dynamic system.

However, there is no warranty that all solution will converge to the steady state (see the following graph)
III. The Solution Path and the Steady State

IV. Solving Dynamic System

Using graphic technique in one dimensional system
IV. Solving Dynamic System

Another possibility to solve dynamic system is to use the computer simulation. An example is to use Excel for simulation (will be introduced in the class)
III. The Solution Path and the Steady State

The steady state (continued)

The steady state has the property that if the solution path of the system is convergent, it must be converge to the steady state (see the following graph)

III. The Solution Path and the Steady State

The solution path

The solution of a system describe how the variables change over time given the initial condition (X0, Y0, …,Z0).
X t f ( X t 1 , Yt 1 , L , Z t 1 ) Yt g ( X t 1 , Yt 1 , L , Z t 1 ) M Z t h ( X t 1 , Yt 1 , L , Z t 1 )
II. The Discrete Time Dynamic System
V. Important Notes
In this course, we generally assume the system is stable, that is, the solution will converge to the steady state. Sometimes, we even ignore the time subscript. In this case, our analysis can be regarded as steady state analysis, that is, we assume that all the variables be at their steady states.
where aij and bi (i,j = 1,2,…,n) are all the parameters.
II. The Discrete Time Dynamic System

The standard form of dynamic system
Note that (3.1) and (3.2) can be regarded as a standard form of discrete dynamic system. Other forms of dynamic system can be transformed into the standard form (examples are provided in the textbook) Many theorems (propositions) to resolve the dynamic system is based on the standard form.
III. The Solution Path and the Steady State

The solution path (continued)

The graphic representation of solution paths
Y
X
t
X
III. The Solution Path and the Steady State

The steady state

The steady state of system (3.1) or (3.2), denoted as (X*, Y*, …, Z*), can be obtained by posing the restriction:
X t X t 1 X * Yt Yt 1 Y * Z t Z t 1 Z *
Macroeconomics
Lecture 3
ቤተ መጻሕፍቲ ባይዱ
Discrete Time Dynamic System: a Technical Preparation
I.
Why Dynamic Analysis is Important?
We have known that macroeconomics is about how macroeconomic variables are determined. In macroeconomics, such a determination can often be described by a dynamic system in terms of either discrete time or continuous time.

If function f(· ), g(· ), …, h(· ) are all linear, the above system may be written as
X t a11 X t 1 a12Yt 1 L a1n Z t 1 b1 Yt a21 X t 1 a22Yt 1 L a2 n Z t 1 b2 M Z t an1 X t 1 an 2Yt 1 L ann Z t 1 bn

IV. Solving Dynamic System


It is not always possible to solve dynamic system, that is, obtaining the solution path. However, in many cases, it is sufficient to detect the stability of the dynamic system, that is, whether the system is convergent to the steady state or not. Such detection can often rely on graphic technique or some well-known mathematic propositions.
III. The Solution Path and the Steady State
Converging to the Steady State
X Y
X* Y*
t
X*
X
III. The Solution Path and the Steady State

The steady state (continued)