Consider the problem

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

with $\displaystyle a_1=0$ is given. $\displaystyle c_t$ is consumption in period t. u(c) is strictly concave and increasing. Consumers can save through a bond. $\displaystyle a_{t+1}$ is the amount of bond accumulated from period t to period t+1.

Solve for the consumer's max problem and express $\displaystyle c_t$ as a function of exogenous variables.

I first solve for the consumers life time budget constraint, which is $\displaystyle \Sigma \frac{c_t}{(1+r)^{t-1}}=\frac {w(1+r)}{r}$

Then i set Lagrange and got the foc :

$\displaystyle \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