Linear First Order Difference Equations (Iterative/General Method)

  1. 1. The problem statement, all variables and given/known data
    I am almost done with a chapter all about this topic and this type of question is the only one I can't get. This is linear first order difference equations. The question is:

    Given the unemployment Ut equation:

    Ut = [tex]\alpha[/tex] + [tex]\beta[/tex] Ut-1
    [tex]\alpha[/tex], [tex]\beta[/tex] > 0

    b. Suppose that there are occasional shocks to the demand for labor causing shifts in Ut. The modified equation for Ut becomes:

    Ut = [tex]\alpha[/tex] + [tex]\beta[/tex] Ut-1 + et

    where et varies over time. Show that the solution to the modified equation is:

    Ut = [tex]\beta[/tex]tU0 + [tex]\frac{\alpha(1-\betat)}{1-\beta}[/tex] + e1[tex]\beta[/tex]t-1 + e2[tex]\beta[/tex]t-2 + ... + et-1[tex]\beta[/tex] + et

    Don't know how to fix that there. It should be (1-[tex]\beta[/tex]t)

    2. Relevant equations

    General Method:
    Pc + Pp = General method

    Yt = (Y0 - [tex]\frac{c}{1+a}[/tex])(-a)t + [tex]\frac{c}{1+a}[/tex]

    I've also got the derived formula for supply and demand but that requires two functions.

    3. The attempt at a solution
    Ok, I can't get the iteration. This is what I've tried:

    Ut = [tex]\alpha[/tex] + [tex]\beta[/tex]Ut-1 + et

    Ut+1 = [tex]\alpha[/tex] + [tex]\beta[/tex]Ut + et+1

    After this point I don't know what to do. I tried to do this:

    Ut+1 = [tex]\beta[/tex]([tex]\alpha[/tex] + [tex]\beta[/tex]Ut + et+1) + [tex]\alpha[/tex] + et+1

    Basically multiplying the whole equation by [tex]\beta[/tex] then adding: [tex]\alpha[/tex] + et+1. Once I do it for 3 periods I can determine the general function but it is different from the given one. I lack the 1-[tex]\beta[/tex] on that denominator. I can solve other equations but have trouble when something else, such as et+1 is added. I've also used the general method but it also turns out different. I was under the impression that I can use the iterative and general solutions for any first order linear difference equation. Am I wrong?

    Any help would be greatly appreciated. Thanks!

    P.S. I would like to thank the system for auto-logging me out while trying to preview my first ever post, thereby deleting a chunk of what I wrote. Good thing I saved. :\
  2. jcsd
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