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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.

The task is attached, so please first read it.

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(t−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
File Type: jpg Samuel.jpg (17.8 KB, 7 views)

Last edited by skipjack; October 21st, 2017 at 02:39 AM.
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October 21st, 2017, 09:52 AM   #2
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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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