Intertemporal choice with possibility of death Here is the setup: 
Suppose that there is an individual who lives up to two periods. He lives with absolute certainty during period $1$, and during this period his sub-utility function is given by:
$$u(c_1) = \log c_1.$$
The individual is alive in period $2$ with probability $p$, and if he is alive, his state sub-utility function is given by
$$u(c_2) =\log c_2$$
and if he dies, he receives the fixed utility level $B$. Suppose he has access to a perfect capital market and discounts the future at rate $\beta$. His endowment is given by $(\omega_1, \omega_2)$.

I have a few questions about how to approach this problem. The first question is: do we assume he can die with debt/excess savings at the end of period $1$? If I assume that $p=0$ (dies with certainty), I get that he takes out a loan $s= \frac{\omega_2}{1+r}$ in the first period. Does this make sense, i.e., do we assume that even if he dies he gets an endowment of $\omega_2$ to pay off his debts? This result would correspond to $c_2 =0$. 

The second question is about the fixed utility level. Should this be interpreted as, if he dies, he gets utility $u(c_1) + B$? Or $u(c_1) + \beta B$? Or that we assume his total lifetime utility from consumption is $u(c_1) = B$? 

Thanks!
It is safe to assume that if the consumer passes away, his creditors get his endowment. This actually happens in real life. $B$ should obviously be zero. As far as the second period is concerned you can represent it as a von Neuman-Morgenstern utility from the lottery:

$\beta u(c_2)$ with probability $p$ and $0$ with probability $1-p$.

I.e. $U(c_1, c_2) = log(c_1) + p\beta log(c_2).$