My Math Forum Samuelson Accelerator Model

 Differential Equations Ordinary and Partial Differential Equations Math Forum

October 21st, 2017, 01:19 AM   #1
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Samuelson Accelerator Model

Hello forum members,
I have an exercise in which I don't know how to handle the derivation. Maybe some experienced members can help me solve this issue.

My first tries to derivate look like this:
This is the source function (in my opinion):
Yt = Ct + It

Transformed it looks like that (in my opinion) (the t-1 is written like an interval in lowered form, like in the Question attached!!):
Ct + It =(Ca+ cyY(t-1)) + (Ia+ vY(t-1))

When I do now the derivation to make it non-homogeneous, it looks like that:
2=(1+ cy(t-1)) + (1+y(tâˆ’1))

But I don't know whether this is correct and how to derivate this (t-1) which looks like a function or a value.

I hope some of You can help me to understand this topic.

Thank You
Cheers!
Attached Images
 Samuel.jpg (17.8 KB, 7 views)

Last edited by skipjack; October 21st, 2017 at 02:39 AM.

 October 21st, 2017, 09:52 AM #2 Senior Member   Joined: May 2016 From: USA Posts: 1,187 Thanks: 489 I'd like to help here, but I am not sure I can. First, you posted this under differential equations, but the text of your problem is talking about difference equations, not differential equations. They are not the same thing. You talk about a "derivation," but I suspect you really mean a "derivative." Second, my iPad makes your attached picture too small to read easily. Can you please copy it out exactly so I can read it. Third, $Y_{t-1}$ simply means the national income for the period immediately preceding period t: the subscripts in t represent indices of DISCRETE time. In other words, $Y_t$ and $Y_{t-1}$ represent different variables, not the same variable performing some mathematical operation on t and t-1. I suspect that what you are being asked to do is to find the equation that describes $\Delta Y$ in terms of $\Delta C$ and $\Delta I$ or else $Y_t$ in terms of $Y_{t-1}.$ Does your text give an explanation of what form of answer is required? Now here comes the final reason that I am reluctant even to try to help: non-homogenous difference equation represent a topic I have not studied. However, once we understand what the problem requires, someone else here can address the lack of homogeniety. Oh, how did you get the equation 2 = a bunch of stuff? Thanks from topsquark and MayWay Last edited by JeffM1; October 21st, 2017 at 09:55 AM.

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