Linear First Order Difference Equations (Iterative/General Method)

rabbitchannel

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

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

U_t = [tex]\beta[/tex]^tU₀ + [tex]\frac{\alpha(1-\beta^t)}{1-\beta}[/tex] + e₁[tex]\beta[/tex]^t-1 + e₂[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 = (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:

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 as e_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. :\