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Math Help - Economics Problem Help

  1. #1
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    Economics Problem Help

    See Attachment. I am the TA for a macroeconomics course. My professor provided me with the answer for this problem (3a.) but I am having a difficult time deriving it using the partial derivatives of the current period C(t) and future period C(t+1). The Euler equation format she has taught the students to use is u'(c1)= (1+r)B*u'(c2) where B is the discount rate and r is the interest rate. Please help!!

    I have been finding the derivative of t=1 and t=2 and then trying to plug these into the Euler equation. For t=1 I get Ct^(-omega) and for t=2 I get B(Ct+1)^(-omega). Plugging these into the Euler equation gives me a B^2 term, which I know is not correct.
    Attached Thumbnails Attached Thumbnails Economics Problem Help-screen-shot-2012-09-30-10.14.22-pm.png  
    Last edited by nyson13; September 30th 2012 at 07:17 PM.
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  2. #2
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    Re: Economics Problem Help

    Hey nyson13.

    Just to clarify, is c2 = c1 + 1? I'll assume this is the case right now (from the context of your post).

    The basic Euler equation is u(x+a) = u(a) + u'(a)(x-a), but you actually have two derivative terms so I'll go off the basis that you are going to use two Euler expansions and then collect the terms. Let a be c1 and b be c2 where u(x+b) = u(b) + u'(b)(x-b). (These are of course approximations not real equalities). Take (2) - (1) and we get:

    u(x+b) - u(x+a) = u(b) - u(a) + u'(b)(x-b) - u'(a)(x-a).
    = u(b) - u(a) + x*(u'(b) - u'(a)) + u'(a)*a - u'(b)*b

    if b = a + 1 then this means

    = u(a+1) - u(a) + x*(u'(a+1) - u'(a)) + u'(a)*a - u'(a+1)*(a+1)

    If x = 0 we get cancellations:

    u(a+1) - u(a) = u(a+1) - u(a) + 0 + u'(a)*a - u'(a+1)*(a+1) which gives
    u'(a)*a - u'(a+1)*(a+1) = 0 which gives

    u'(a)*a = (1+a)u'(a+1) or
    u'(a) = [1+a]/a * u'(a+1) or
    u'(a) = [1/a + 1]*u'(a+1) with the Euler approximation.

    Now do you have values for u'(a) and for u'(a+1) in terms of u's and a's (your notation is a little confusing) and does it match this? (Or have I screwed up somewhere along the lines?)
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