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

by rabbitchannel
Tags: difference, equations, iterative or general, linear, method, order
 P: 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 = $$\alpha$$ + $$\beta$$ Ut-1 $$\alpha$$, $$\beta$$ > 0 b. Suppose that there are occasional shocks to the demand for labor causing shifts in Ut. The modified equation for Ut becomes: Ut = $$\alpha$$ + $$\beta$$ Ut-1 + et where et varies over time. Show that the solution to the modified equation is: Ut = $$\beta$$tU0 + $$\frac{\alpha(1-\betat)}{1-\beta}$$ + e1$$\beta$$t-1 + e2$$\beta$$t-2 + ... + et-1$$\beta$$ + et Don't know how to fix that there. It should be (1-$$\beta$$t) 2. Relevant equations General Method: Pc + Pp = General method Yt = (Y0 - $$\frac{c}{1+a}$$)(-a)t + $$\frac{c}{1+a}$$ 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 = $$\alpha$$ + $$\beta$$Ut-1 + et Ut+1 = $$\alpha$$ + $$\beta$$Ut + et+1 After this point I don't know what to do. I tried to do this: Ut+1 = $$\beta$$($$\alpha$$ + $$\beta$$Ut + et+1) + $$\alpha$$ + et+1 Basically multiplying the whole equation by $$\beta$$ then adding: $$\alpha$$ + 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-$$\beta$$ 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. :\

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