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Math Help - macro economic maximization problem

  1. #1
    Junior Member
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    macro economic maximization problem

    Consider the problem
     max(c_t,a_{t+1}) \Sigma B^{t-1}U(c_t) , where U(c)=log(c-b) subject to  c_t+a_{t+1}=(1+r)a_t + w

    with a_1=0 is given. c_t is consumption in period t. u(c) is strictly concave and increasing. Consumers can save through a bond. a_{t+1} is the amount of bond accumulated from period t to period t+1.
    Solve for the consumer's max problem and express c_t as a function of exogenous variables.

    I first solve for the consumers life time budget constraint, which is \Sigma \frac{c_t}{(1+r)^{t-1}}=\frac {w(1+r)}{r}
    Then i set Lagrange and got the foc :
    \frac {B^{t-1}}{c_t-b} = \frac {\lambda}{(1+r)^{t-1}}

    But now i am unsure how to continue

    Any help is appreciated.
    Thanks in advance
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  2. #2
    MHF Contributor
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    I looked at this for a while and cant get a complete solution, but since no one replied for 4 days ill point out that if you divide your FOC for c(t) by the FOC for c(t-1) you get the relationship between c(t) and c(t-1).

    Then you need to choose the value of c(1) that makes the whole progression fit within the budget constraint, although i couldn't work out how...so i may be pointing you in the wrong direction.
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