I am trying to mathematically represent expected discounted utilities when the length of a lifetime is uncertain.
$V_0$ is the expected discounted utility at $t=0$ and can be represented as such:
$V_0=\displaystyle \sum_{t=0}^{T}[1-F(t-1)]\beta^tu(c_t)$
*$F(t)=P(\hat T\leq t)$ and $f(t)=P(\hat T=t)$, where $\hat T$ is some random age at which the person dies.
I need to show that the recursive representation for expected discounted utilities is:
$V_t=u(c_t)+\beta\frac{1-F(t)}{1-F(t-1)}V_{t+1}$
I have been trying to prove this to myself through comparing $V_1$ and $V_0$ but have been having some difficulty doing so.
Expanded, $V_0$ is $V_0=[1-F(-1)]u(c_0)+[1-F(0)]\beta u(c_1)+ \cdots +[1-F(T-1)]\beta^Tu(c_T)$.
I think that $V_1$ should be: $V_1=[1-F(0)]u(c_1)+[1-F(1)]\beta u(c_2)+[1-F(2)]\beta^2 u(c_3)+ \cdots+[1-F(T-1)]\beta^{T-1}u(c_T)$
as the discount factors should change (since we are one period ahead in the future), but I'm not quite sure about how the probabilities should change. Do I need to include conditional probabilities?
Another way: I also tried to understand the equation by representing the recursive equation as $V_{t+1}$ in terms of $V_t$.
$V_{t+1}=(V_t-u(c_t))\cdot \beta^{-1}\cdot\frac{1-F(t-1)}{1-F(t)}$
However, when I tried using this equation to expand $V_1$, I got a bit confused, as I ended up with:
$V_1=[1-F(-1)]u(c_1)+\frac{[1-F(1)][1-F(-1)]}{1-F(0)}\beta u(c_2)+ \cdots +\frac{[1-F(T-1)][1-F(-1)]}{1-F(0)}\beta^{T-1}u(c_T)$
which doesn't really make any sense to me.
What am I doing wrong? Am I understanding $V_1$ or $V_0$ in a completely incorrect way?