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Linear First Order Difference Equations (Iterative/General Method) 
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#1
Mar2110, 05:46 PM

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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_{t1} [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_{t1} + e_{t} where e_{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]^{t1} + e_{2}[tex]\beta[/tex]^{t2} + ... + e_{t1}[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_{t1} + 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 autologging 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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