macro economic maximization problem

**firebio** · February 12th 2011, 04:43 PM

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

**SpringFan25** · February 17th 2011, 11:27 AM

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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