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Math Help - Forming first-order difference equation

  1. #1
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    Forming first-order difference equation

    Hi,

    I was wondering if anyone can help with this mathematical economics problem:


    The multiplier-accelerator model of growth has the following three equations


    St = aYt
    It+1 = b(Yt+1 - Yt)
    St = It


    a) Briefly interpret the above three equations and combine them as a first-order difference equation.


    b) Assume that output at time zero is Y0 and solve the difference equation. Which are the determinants of output growth in the model? Explain intuitively.



    It's clear from b) that the difference equation must refer to Y(t). But I can't work out how to get a single first-order difference equation that includes the three equations, that is set up to refer to Y(t).
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  2. #2
    MHF Contributor

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    Re: Forming first-order difference equation

    I assume "Yt+1" is Y_{t+1}, not " Y_t+ 1" and that " It+1" is I_{t+1}. So the middle equation is the difference equation: Y_{t+1}- Y_t= \frac{I_{t+1}}{b}

    Since you are given that I_t= S_t and S_t= aY_t, it follows that I_t= aY_t so that I_{t+1}= aY_{t+1}. Putting that into your equation, Y_{t+1}- Y_t= \frac{a}{b}Y_{t+1} which is the same as \left(1- \frac{a}{b}\right)Y_{t+1}= Y_t. I would write that as Y_{t+1}= \frac{b}{a- b}Y_t is relatively easy to solve.
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  3. #3
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    Re: Forming first-order difference equation

    Excellent, thanks
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