Optimal spending over several periods with log utility and uncertain lifetime

Question

If someone has probability p(n) of being alive after n periods and p(n) is known with p(n) = 0 for n >= m, and if he has log utility of consumption, and his utilities are additive over time, and interest rates and inflation are zero, can one derive the fraction of wealth he should consume in each period? I think that the amount of wealth he sets aside for a period is proportional to the probability he will be alive then but have not proven it.

I have solved the two-period case. If he starts with one unit of wealth, and he has probability p of living to a second period, and no probability of living after that, and he has log utility of spending, then his total utility is

U(x) = log(x) + p*log(1-x)

U'(x) = 1/x - p/(1-x)

Setting U'(x) = 0 gives

p*x = 1-x

So

x = 1/(1+p)

which makes sense. If p = 1 he is guaranteed to be alive in the 2nd time period, and he splits consumption equally between the two time periods. If p = 0 he spends all his wealth in the first period.

Answer 1

Hint

Generalizing from your 2-period exercise, let $N=m-1$ be the last possible period of being alive. The problem to solve is \begin{equation} \max_{\{c_n\}_{n=0}^N}\;u(c_0)+p(1)u(c_1)+\cdots+p(N)u(c_N) =\max_{\{c_n\}_{n=0}^N}\;\sum_{n=0}^Np(n)\log(c_n) \end{equation} subject to a lifetime budget constraint: $c_0++c_1+\cdots+c_N=y$ where $y$ is the lifetime income.