Lecture 3 Technique Preparation Discrete Time Dynamic System(宏观经济学(南开大学,龚刚)

1、Lecture 3Discrete Time Dynamic System: a Technical PreparationMacroeconomicsMacroeconomicsI.Why Dynamic Analysis is Important?nWe have known that macroeconomics is about how macroeconomic variables are determined.nIn macroeconomics, such a determination can often be described by a dynamic system in

2、terms of either discrete time or continuous time.II. The Discrete Time Dynamic SystemnLet 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: 111111111(,)(,)(,)ttttttttttttXfXYZYgXY

3、ZZhXYZLLMLII. The Discrete Time Dynamic SystemnIf function f(), g(), , h() are all linear, the above system may be written as where aij and bi (i,j = 1,2,n) are all the parameters.11112111121122121211211tttnttttnttntntnntnXaXaYaZbYaXaYaZbZaXaYaZbLLMLII. The Discrete Time Dynamic SystemnThe standard

4、form of dynamic systemnNote that (3.1) and (3.2) can be regarded as a standard form of discrete dynamic system. nOther forms of dynamic system can be transformed into the standard form (examples are provided in the textbook)nMany theorems (propositions) to resolve the dynamic system is based on the

5、standard form.III. The Solution Path and the Steady StatenThe solution pathnThe solution of a system describe how the variables change over time given the initial condition (X0, Y0, ,Z0).III. The Solution Path and the Steady StatenThe solution path (continued)nThe graphic representation of solution

6、paths tXXYIII. The Solution Path and the Steady StatenThe steady state nThe steady state of system (3.1) or (3.2), denoted as (X*, Y*, , Z*), can be obtained by posing the restriction: *1*1*1ttttttXXXYYYZZZIII. The Solution Path and the Steady StatenThe steady state (continued)nThe steady state has

7、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 StateConverging to the Steady StatetXXYX*X*Y*III. The Solution Path and the Steady StatenThe steady state (continued)nHowever, th

8、ere is no warranty that all solution will converge to the steady state (see the following graph)III. The Solution Path and the Steady StateNo Converging to the Steady StatetXXYX*X*Y*III. The Solution Path and the Steady StatenThe steady state (continued)nThere is no warranty that the steady state is

9、 unique (multiple steady states could occur)nThere is also no warranty that the steady state even exists (it could be complex). nBoth of the above are more likely to occur in a nonlinear dynamic system.IV. Solving Dynamic SystemnIt is not always possible to solve dynamic system, that is, obtaining t

10、he solution path. nHowever, 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.nSuch detection can often rely on graphic technique or some well-known mathematic propositions. IV. Solving Dynamic SystemnU

11、sing graphic technique in one dimensional systemIV. Solving Dynamic SystemnAnother 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)V. Important NotesnIn this course, we generally assume the system is stable, that is, the solution will converge to the steady state.nSometi