(adsbygoogle = window.adsbygoogle || []).push({}); 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 U_{t}equation:

U_{t}= [tex]\alpha[/tex] + [tex]\beta[/tex] U_{t-1}

[tex]\alpha[/tex], [tex]\beta[/tex] > 0

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

U_{t}= [tex]\alpha[/tex] + [tex]\beta[/tex] U_{t-1}+e_{t}

wheree_{t}varies over time. Show that the solution to the modified equation is:

U_{t}= [tex]\beta[/tex]^{t}U_{0}+ [tex]\frac{\alpha(1-\beta^{t})}{1-\beta}[/tex] +e_{1}[tex]\beta[/tex]^{t-1}+e_{2}[tex]\beta[/tex]^{t-2}+ ... +e_{t-1}[tex]\beta[/tex] +e_{t}

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

2. Relevant equations

General Method:

P_{c}+ P_{p}= General method

Y_{t}= (Y_{0}- [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:

U_{t}= [tex]\alpha[/tex] + [tex]\beta[/tex]U_{t-1}+e_{t}

U_{t+1}= [tex]\alpha[/tex] + [tex]\beta[/tex]U_{t}+e_{t+1}

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

U_{t+1}= [tex]\beta[/tex]([tex]\alpha[/tex] + [tex]\beta[/tex]U_{t}+e_{t+1}) + [tex]\alpha[/tex] +e_{t+1}

Basically multiplying the whole equation by [tex]\beta[/tex] then adding: [tex]\alpha[/tex] +e_{t+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 ase_{t+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. :\

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# Linear First Order Difference Equations (Iterative/General Method)

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