I am trying to solve this finite horizon dynamic problem (consumption-savings) using backward induction.
Maximize $\sum_{t=0}^{T}u(c_{t})$
subject to $w_{0}>0, c\in [0,w],w(t+1)=(w_{t}-c_{t})(1+r)$
and $u(c_{t})=c^{\alpha},\alpha\in(0,1)$
where, u=utility from consumption, w=wealth of consumer, r=rate of interest, c=consumption.
I have solved this question so far.
Maximize the value function at each stage: $c^{\alpha}+V((w-c)(1+r))$
Backward induction gives optimal strategy at t=T-2:
Optimal strategy=$c_{T-2}(w)=\dfrac{w(1+(1+r)^{2\alpha})^{1/(\alpha-1)}(r+2)}{1+r+(1+(1+r)^{2\alpha})^{1/(\alpha-1)}(2+r)}$
If I plug it into the value function to get the maximized value function, I get a complicated expression which I am not able to solve to get the general form of value function across all periods.
Am I doing anything wrong or is there any simpler way to solve this? Would be grateful for any hint regarding this.